Theorem isofval2 50109

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 17943	. . . . 5 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
3		isofval2.i	. . . . . . 7 ⊢ 𝐼 = (Iso‘𝐶)
4	3	fneq1i 6634	. . . . . 6 ⊢ (𝐼 Fn (𝐵 × 𝐵) ↔ (Iso‘𝐶) Fn (𝐵 × 𝐵))
5		isofval2.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
6	5, 5	xpeq12i 5679	. . . . . . 7 ⊢ (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶))
7	6	fneq2i 6635	. . . . . 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 7549	. . . 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 17933	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥𝐼𝑦) = dom (𝑥𝑁𝑦))
18	17	mpoeq3dva 7495	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐼𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥𝑁𝑦)))
19	12, 18	eqtrd 2796	1 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥𝑁𝑦)))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   × cxp 5649  dom cdm 5651   Fn wfn 6532  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420  Basecbs 17380  Catccat 17831  Invcinv 17913  Isociso 17914

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-inv 17916  df-iso 17917

This theorem is used by:  isorcl2  50111  isopropdlem  50117