Theorem cic

Description: Objects X and Y in a category are isomorphic provided that there is an isomorphism f : X --> Y , see definition 3.15 of Adamek p. 29. (Contributed by AV, 4-Apr-2020)

	Ref	Expression
Hypotheses	cic.i	$$⊢I=\mathrm{Iso}\left(C\right)$$
	cic.b	$$⊢B={\mathrm{Base}}_C$$
	cic.c	$$⊢\phi \to C\in \mathrm{Cat}$$
	cic.x	$$⊢\phi \to X\in B$$
	cic.y	$$⊢\phi \to Y\in B$$
Assertion	cic	$$⊢\phi \to \left(X{\simeq }_{𝑐}\left(C\right)Y\leftrightarrow \exists ff\in \left(XIY\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		cic.i	$$⊢I=\mathrm{Iso}\left(C\right)$$
2		cic.b	$$⊢B={\mathrm{Base}}_C$$
3		cic.c	$$⊢\phi \to C\in \mathrm{Cat}$$
4		cic.x	$$⊢\phi \to X\in B$$
5		cic.y	$$⊢\phi \to Y\in B$$
6	1 2 3 4 5	brcic	$$⊢\phi \to \left(X{\simeq }_{𝑐}\left(C\right)Y\leftrightarrow XIY\ne \varnothing \right)$$
7		n0	$$⊢XIY\ne \varnothing \leftrightarrow \exists ff\in \left(XIY\right)$$
8	6 7	bitrdi	$$⊢\phi \to \left(X{\simeq }_{𝑐}\left(C\right)Y\leftrightarrow \exists ff\in \left(XIY\right)\right)$$