Theorem isofval2 49650

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 17808	. . . . 5 ⊢ (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
3		isofval2.i	. . . . . . 7 ⊢ 𝐼 = (Iso‘𝐶)
4	3	fneq1i 6618	. . . . . 6 ⊢ (𝐼 Fn (𝐵 × 𝐵) ↔ (Iso‘𝐶) Fn (𝐵 × 𝐵))
5		isofval2.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
6	5, 5	xpeq12i 5675	. . . . . . 7 ⊢ (𝐵 × 𝐵) = ((Base‘𝐶) × (Base‘𝐶))
7	6	fneq2i 6619	. . . . . 6 ⊢ ((Iso‘𝐶) Fn (𝐵 × 𝐵) ↔ (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
8	4, 7	bitri 277	. . . . 5 ⊢ (𝐼 Fn (𝐵 × 𝐵) ↔ (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
9	2, 8	sylibr 236	. . . 4 ⊢ (𝐶 ∈ Cat → 𝐼 Fn (𝐵 × 𝐵))
10		fnov 7527	. . . 4 ⊢ (𝐼 Fn (𝐵 × 𝐵) ↔ 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐼𝑦)))
11	9, 10	sylib 220	. . 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 1146	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Cat)
15		simp2 1150	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)
16		simp3 1151	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
17	5, 13, 14, 15, 16, 3	isoval 17798	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥𝐼𝑦) = dom (𝑥𝑁𝑦))
18	17	mpoeq3dva 7473	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐼𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥𝑁𝑦)))
19	12, 18	eqtrd 2797	1 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥𝑁𝑦)))

Syntax hints:   → wi 4   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   × cxp 5645  dom cdm 5647   Fn wfn 6516  ‘cfv 6521  (class class class)co 7396   ∈ cmpo 7398  Basecbs 17245  Catccat 17696  Invcinv 17778  Isociso 17779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-inv 17781  df-iso 17782

This theorem is referenced by:  isorcl2  49652  isopropdlem  49658