Theorem isisod 49059

Description: The predicate "is an isomorphism" (deduction form). (Contributed by Zhi Wang, 16-Sep-2025.)

Hypotheses

Ref	Expression
isisod.b	⊢ 𝐵 = (Base‘𝐶)
isisod.h	⊢ 𝐻 = (Hom ‘𝐶)
isisod.o	⊢ · = (comp‘𝐶)
isisod.i	⊢ 𝐼 = (Iso‘𝐶)
isisod.1	⊢ 1 = (Id‘𝐶)
isisod.c	⊢ (𝜑 → 𝐶 ∈ Cat)
isisod.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
isisod.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
isisod.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
isisod.g	⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑋))
isisod.gf	⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋))
isisod.fg	⊢ (𝜑 → (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝐺) = ( 1 ‘𝑌))

Assertion

Ref	Expression
isisod	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))

Proof of Theorem isisod

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isisod.gf	. . 3 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋))
2		isisod.fg	. . 3 ⊢ (𝜑 → (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝐺) = ( 1 ‘𝑌))
3		isisod.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑋))
4		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
5	4	oveq1d 7356	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹))
6	5	eqeq1d 2733	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → ((𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋) ↔ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋)))
7	4	oveq2d 7357	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝑔) = (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝐺))
8	7	eqeq1d 2733	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → ((𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝑔) = ( 1 ‘𝑌) ↔ (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝐺) = ( 1 ‘𝑌)))
9	6, 8	anbi12d 632	. . . 4 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (((𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝑔) = ( 1 ‘𝑌)) ↔ ((𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝐺) = ( 1 ‘𝑌))))
10	3, 9	rspcedv 3565	. . 3 ⊢ (𝜑 → (((𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝐺) = ( 1 ‘𝑌)) → ∃𝑔 ∈ (𝑌𝐻𝑋)((𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝑔) = ( 1 ‘𝑌))))
11	1, 2, 10	mp2and 699	. 2 ⊢ (𝜑 → ∃𝑔 ∈ (𝑌𝐻𝑋)((𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝑔) = ( 1 ‘𝑌)))
12		isisod.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
13		isisod.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
14		isisod.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
15		isisod.i	. . 3 ⊢ 𝐼 = (Iso‘𝐶)
16		isisod.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
17		isisod.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
18		isisod.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
19		isisod.1	. . 3 ⊢ 1 = (Id‘𝐶)
20		isisod.o	. . . 4 ⊢ · = (comp‘𝐶)
21	20	oveqi 7354	. . 3 ⊢ (⟨𝑋, 𝑌⟩ · 𝑋) = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
22	20	oveqi 7354	. . 3 ⊢ (⟨𝑌, 𝑋⟩ · 𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
23	12, 13, 14, 15, 16, 17, 18, 19, 21, 22	dfiso2 17674	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)((𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝑔) = ( 1 ‘𝑌))))
24	11, 23	mpbird 257	1 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  ⟨cop 4577  ‘cfv 6476  (class class class)co 7341  Basecbs 17115  Hom chom 17167  compcco 17168  Catccat 17565  Idccid 17566  Isociso 17648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-sect 17649  df-inv 17650  df-iso 17651

This theorem is referenced by:  upciclem4  49201