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Metamath Proof Explorer

Theorem idresperm

Description: The identity function restricted to a set is a permutation of this set. (Contributed by AV, 17-Mar-2019)

Ref Expression
Hypothesis idresperm.g ⊢ G = SymGrp ⁡ A
Assertion idresperm ⊢ A ∈ V → I ↾ A ∈ Base G

Proof

Step Hyp Ref Expression
1 idresperm.g ⊢ G = SymGrp ⁡ A
2 f1oi ⊢ I ↾ A : A ⟶ 1-1 onto A
3 eqid ⊢ Base G = Base G
4 1 3 elsymgbas ⊢ A ∈ V → I ↾ A ∈ Base G ↔ I ↾ A : A ⟶ 1-1 onto A
5 2 4 mpbiri ⊢ A ∈ V → I ↾ A ∈ Base G