Theorem 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	$$⊢B={\mathrm{Base}}_C$$
	invfval.n	$$⊢N=\mathrm{Inv}\left(C\right)$$
	invfval.c	$$⊢\phi \to C\in \mathrm{Cat}$$
	invss.x	$$⊢\phi \to X\in B$$
	invss.y	$$⊢\phi \to Y\in B$$
	isoval.n	$$⊢I=\mathrm{Iso}\left(C\right)$$
Assertion	invf1o	$$⊢\phi \to \left(XNY\right):XIY\underset{1-1\; onto}{⟶}YIX$$

Proof

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