Theorem isofval2

Description: Function value of the function returning the isomorphisms of a category. (Contributed by Zhi Wang, 27-Oct-2025)

	Ref	Expression
Hypotheses	isofval2.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	isofval2.n	⊢ 𝑁 = ( Inv ‘ 𝐶 )
	isofval2.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	isofval2.i	⊢ 𝐼 = ( Iso ‘ 𝐶 )
Assertion	isofval2	⊢ ( 𝜑 → 𝐼 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ dom ( 𝑥 𝑁 𝑦 ) ) )

Proof

Step	Hyp	Ref	Expression
1		isofval2.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
2		isofval2.n	⊢ 𝑁 = ( Inv ‘ 𝐶 )
3		isofval2.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
4		isofval2.i	⊢ 𝐼 = ( Iso ‘ 𝐶 )
5		isofn	⊢ ( 𝐶 ∈ Cat → ( Iso ‘ 𝐶 ) Fn ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) )
6	4	fneq1i	⊢ ( 𝐼 Fn ( 𝐵 × 𝐵 ) ↔ ( Iso ‘ 𝐶 ) Fn ( 𝐵 × 𝐵 ) )
7	1 1	xpeq12i	⊢ ( 𝐵 × 𝐵 ) = ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) )
8	7	fneq2i	⊢ ( ( Iso ‘ 𝐶 ) Fn ( 𝐵 × 𝐵 ) ↔ ( Iso ‘ 𝐶 ) Fn ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) )
9	6 8	bitri	⊢ ( 𝐼 Fn ( 𝐵 × 𝐵 ) ↔ ( Iso ‘ 𝐶 ) Fn ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) )
10	5 9	sylibr	⊢ ( 𝐶 ∈ Cat → 𝐼 Fn ( 𝐵 × 𝐵 ) )
11		fnov	⊢ ( 𝐼 Fn ( 𝐵 × 𝐵 ) ↔ 𝐼 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 𝐼 𝑦 ) ) )
12	10 11	sylib	⊢ ( 𝐶 ∈ Cat → 𝐼 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 𝐼 𝑦 ) ) )
13	3 12	syl	⊢ ( 𝜑 → 𝐼 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 𝐼 𝑦 ) ) )
14	3	3ad2ant1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → 𝐶 ∈ Cat )
15		simp2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 )
16		simp3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → 𝑦 ∈ 𝐵 )
17	1 2 14 15 16 4	isoval	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 𝐼 𝑦 ) = dom ( 𝑥 𝑁 𝑦 ) )
18	17	mpoeq3dva	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 𝐼 𝑦 ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ dom ( 𝑥 𝑁 𝑦 ) ) )
19	13 18	eqtrd	⊢ ( 𝜑 → 𝐼 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ dom ( 𝑥 𝑁 𝑦 ) ) )