Theorem oppciso

Description: An isomorphism in the opposite category. See also remark 3.9 in Adamek p. 28. (Contributed by Mario Carneiro, 3-Jan-2017)

	Ref	Expression
Hypotheses	oppcsect.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	oppcsect.o	⊢ 𝑂 = ( oppCat ‘ 𝐶 )
	oppcsect.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	oppcsect.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	oppcsect.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	oppciso.s	⊢ 𝐼 = ( Iso ‘ 𝐶 )
	oppciso.t	⊢ 𝐽 = ( Iso ‘ 𝑂 )
Assertion	oppciso	⊢ ( 𝜑 → ( 𝑋 𝐽 𝑌 ) = ( 𝑌 𝐼 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		oppcsect.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
2		oppcsect.o	⊢ 𝑂 = ( oppCat ‘ 𝐶 )
3		oppcsect.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
4		oppcsect.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		oppcsect.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6		oppciso.s	⊢ 𝐼 = ( Iso ‘ 𝐶 )
7		oppciso.t	⊢ 𝐽 = ( Iso ‘ 𝑂 )
8		eqid	⊢ ( Inv ‘ 𝐶 ) = ( Inv ‘ 𝐶 )
9		eqid	⊢ ( Inv ‘ 𝑂 ) = ( Inv ‘ 𝑂 )
10	1 2 3 4 5 8 9	oppcinv	⊢ ( 𝜑 → ( 𝑋 ( Inv ‘ 𝑂 ) 𝑌 ) = ( 𝑌 ( Inv ‘ 𝐶 ) 𝑋 ) )
11	10	dmeqd	⊢ ( 𝜑 → dom ( 𝑋 ( Inv ‘ 𝑂 ) 𝑌 ) = dom ( 𝑌 ( Inv ‘ 𝐶 ) 𝑋 ) )
12	2 1	oppcbas	⊢ 𝐵 = ( Base ‘ 𝑂 )
13	2	oppccat	⊢ ( 𝐶 ∈ Cat → 𝑂 ∈ Cat )
14	3 13	syl	⊢ ( 𝜑 → 𝑂 ∈ Cat )
15	12 9 14 4 5 7	isoval	⊢ ( 𝜑 → ( 𝑋 𝐽 𝑌 ) = dom ( 𝑋 ( Inv ‘ 𝑂 ) 𝑌 ) )
16	1 8 3 5 4 6	isoval	⊢ ( 𝜑 → ( 𝑌 𝐼 𝑋 ) = dom ( 𝑌 ( Inv ‘ 𝐶 ) 𝑋 ) )
17	11 15 16	3eqtr4d	⊢ ( 𝜑 → ( 𝑋 𝐽 𝑌 ) = ( 𝑌 𝐼 𝑋 ) )