invf1o

Description: The inverse relation is a bijection from isomorphisms to isomorphisms. This means that every isomorphism F e. ( X I Y ) has a unique inverse, denoted by ( ( InvC )F ) . Remark 3.12 of Adamek p. 28. (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Hypotheses	invfval.b	$⊢ B = {Base}_{C}$
	invfval.n	$⊢ N = Inv (C)$
	invfval.c	$⊢ φ \to C \in Cat$
	invss.x	$⊢ φ \to X \in B$
	invss.y	$⊢ φ \to Y \in B$
	isoval.n	$⊢ I = Iso (C)$
Assertion	invf1o	$⊢ φ \to (X N Y) : X I Y \underset{1-1 onto}{⟶} Y I X$

Proof

Step	Hyp	Ref	Expression
1		invfval.b	$⊢ B = {Base}_{C}$
2		invfval.n	$⊢ N = Inv (C)$
3		invfval.c	$⊢ φ \to C \in Cat$
4		invss.x	$⊢ φ \to X \in B$
5		invss.y	$⊢ φ \to Y \in B$
6		isoval.n	$⊢ I = Iso (C)$
7	1 2 3 4 5 6	invf	$⊢ φ \to (X N Y) : X I Y ⟶ Y I X$
8	7	ffnd	$⊢ φ \to (X N Y) Fn (X I Y)$
9	1 2 3 5 4 6	invf	$⊢ φ \to (Y N X) : Y I X ⟶ X I Y$
10	9	ffnd	$⊢ φ \to (Y N X) Fn (Y I X)$
11	1 2 3 4 5	invsym2	$⊢ φ \to {(X N Y)}^{-1} = Y N X$
12	11	fneq1d	$⊢ φ \to ({(X N Y)}^{-1} Fn (Y I X) \leftrightarrow (Y N X) Fn (Y I X))$
13	10 12	mpbird	$⊢ φ \to {(X N Y)}^{-1} Fn (Y I X)$
14		dff1o4	$⊢ (X N Y) : X I Y \underset{1-1 onto}{⟶} Y I X \leftrightarrow ((X N Y) Fn (X I Y) \land {(X N Y)}^{-1} Fn (Y I X))$
15	8 13 14	sylanbrc	$⊢ φ \to (X N Y) : X I Y \underset{1-1 onto}{⟶} Y I X$

Metamath Proof Explorer

Theorem invf1o

Proof