Theorem isofval2 48994

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 17713	. . . . 5 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
3		isofval2.i	. . . . . . 7 ⊢ 𝐼 = (Iso‘𝐶)
4	3	fneq1i 6597	. . . . . 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 6598	. . . . . 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 7500	. . . 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 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Cat)
15		simp2 1137	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)
16		simp3 1138	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
17	5, 13, 14, 15, 16, 3	isoval 17703	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥𝐼𝑦) = dom (𝑥𝑁𝑦))
18	17	mpoeq3dva 7446	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐼𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥𝑁𝑦)))
19	12, 18	eqtrd 2764	1 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥𝑁𝑦)))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   × cxp 5629  dom cdm 5631   Fn wfn 6494  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  Basecbs 17155  Catccat 17601  Invcinv 17683  Isociso 17684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  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 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-inv 17686  df-iso 17687

This theorem is referenced by:  isorcl2  48996  isopropdlem  49002