Theorem isofval2 49810

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

Hypotheses

Ref	Expression
isofval2.b	⊢ 𝐵 = (Base‘𝐶)
isofval2.n	⊢ 𝑁 = (Inv‘𝐶)
isofval2.c	⊢ (𝜑 → 𝐶 ∈ Cat)
isofval2.i	⊢ 𝐼 = (Iso‘𝐶)

Assertion

Ref	Expression
isofval2	⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥𝑁𝑦)))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐼,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝑁(𝑥,𝑦)

Proof of Theorem isofval2

Step	Hyp	Ref	Expression
1		isofval2.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
2		isofn 17827	. . . . 5 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
3		isofval2.i	. . . . . . 7 ⊢ 𝐼 = (Iso‘𝐶)
4	3	fneq1i 6632	. . . . . 6 ⊢ (𝐼 Fn (𝐵 × 𝐵) ↔ (Iso‘𝐶) Fn (𝐵 × 𝐵))
5		isofval2.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
6	5, 5	xpeq12i 5689	. . . . . . 7 ⊢ (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶))
7	6	fneq2i 6633	. . . . . 6 ⊢ ((Iso‘𝐶) Fn (𝐵 × 𝐵) ↔ (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
8	4, 7	bitri 278	. . . . 5 ⊢ (𝐼 Fn (𝐵 × 𝐵) ↔ (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
9	2, 8	sylibr 237	. . . 4 ⊢ (𝐶 ∈ Cat → 𝐼 Fn (𝐵 × 𝐵))
10		fnov 7541	. . . 4 ⊢ (𝐼 Fn (𝐵 × 𝐵) ↔ 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐼𝑦)))
11	9, 10	sylib 221	. . 3 ⊢ (𝐶 ∈ Cat → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐼𝑦)))
12	1, 11	syl 18	. 2 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐼𝑦)))
13		isofval2.n	. . . 4 ⊢ 𝑁 = (Inv‘𝐶)
14	1	3ad2ant1 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Cat)
15		simp2 1155	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)
16		simp3 1156	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
17	5, 13, 14, 15, 16, 3	isoval 17817	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥𝐼𝑦) = dom (𝑥𝑁𝑦))
18	17	mpoeq3dva 7487	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐼𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥𝑁𝑦)))
19	12, 18	eqtrd 2798	1 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥𝑁𝑦)))

Syntax hints:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   × cxp 5659  dom cdm 5661   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  Basecbs 17264  Catccat 17715  Invcinv 17797  Isociso 17798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-inv 17800  df-iso 17801

This theorem is referenced by:  isorcl2  49812  isopropdlem  49818