Theorem isofval2 49507

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 17742	. . . . 5 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
3		isofval2.i	. . . . . . 7 ⊢ 𝐼 = (Iso‘𝐶)
4	3	fneq1i 6595	. . . . . 6 ⊢ (𝐼 Fn (𝐵 × 𝐵) ↔ (Iso‘𝐶) Fn (𝐵 × 𝐵))
5		isofval2.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
6	5, 5	xpeq12i 5659	. . . . . . 7 ⊢ (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶))
7	6	fneq2i 6596	. . . . . 6 ⊢ ((Iso‘𝐶) Fn (𝐵 × 𝐵) ↔ (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
8	4, 7	bitri 275	. . . . 5 ⊢ (𝐼 Fn (𝐵 × 𝐵) ↔ (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
9	2, 8	sylibr 234	. . . 4 ⊢ (𝐶 ∈ Cat → 𝐼 Fn (𝐵 × 𝐵))
10		fnov 7498	. . . 4 ⊢ (𝐼 Fn (𝐵 × 𝐵) ↔ 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐼𝑦)))
11	9, 10	sylib 218	. . 3 ⊢ (𝐶 ∈ Cat → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐼𝑦)))
12	1, 11	syl 17	. 2 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐼𝑦)))
13		isofval2.n	. . . 4 ⊢ 𝑁 = (Inv‘𝐶)
14	1	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Cat)
15		simp2 1138	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)
16		simp3 1139	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
17	5, 13, 14, 15, 16, 3	isoval 17732	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥𝐼𝑦) = dom (𝑥𝑁𝑦))
18	17	mpoeq3dva 7444	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐼𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥𝑁𝑦)))
19	12, 18	eqtrd 2771	1 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥𝑁𝑦)))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   × cxp 5629  dom cdm 5631   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  Basecbs 17179  Catccat 17630  Invcinv 17712  Isociso 17713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-inv 17715  df-iso 17716

This theorem is referenced by:  isorcl2  49509  isopropdlem  49515