`BookProof.ChapterGaugeParametrization.isComprehensiveGaugeFixing_iff_surjOn` (π : X → Y) (S : Set X) : IsComprehensiveGaugeFixing (fiberGauge π) S ↔ ∀ x : X, ∃ s ∈ S, π s = π x

Proved

BookProof.ChapterGaugeParametrization.isComprehensiveGaugeFixing_iff_surjOn

by leonardopedro · Oct 8, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterGaugeParametrization.

BookProof.ChapterGaugeParametrization.isComprehensiveGaugeFixing_iff_surjOn (π : X → Y) (S : Set X) : IsComprehensiveGaugeFixing (fiberGauge π) S ↔ ∀ x : X, ∃ s ∈ S, π s = π x

Formalization note: Lean 4 identifier BookProof.ChapterGaugeParametrization.isComprehensiveGaugeFixing_iff_surjOn.

Preamble

-- Generated from ChapterGaugeParametrization.lean — theorem BookProof.ChapterGaugeParametrization.isComprehensiveGaugeFixing_iff_surjOn
import Mathlib
import Definitions.Def_ChapterGaugeParametrization
import Definitions.Def_ChapterGaugeIncompleteFixing
open BookProof.ChapterGaugeIncompleteFixing
open BookProof.ChapterGaugeParametrization



open BookProof.ChapterGaugeIncompleteFixing

variable {X Y : Type*}

Formal statement

theorem BookProof.ChapterGaugeParametrization.isComprehensiveGaugeFixing_iff_surjOn (π : X → Y) (S : Set X) :
    IsComprehensiveGaugeFixing (fiberGauge π) S ↔ ∀ x : X, ∃ s ∈ S, π s = π x := by sorry

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