Theorem isisod 48876

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 7414	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹))
6	5	eqeq1d 2736	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → ((𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋) ↔ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋)))
7	4	oveq2d 7415	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝑔) = (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝐺))
8	7	eqeq1d 2736	. . . . 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 3592	. . 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 7412	. . 3 ⊢ (⟨𝑋, 𝑌⟩ · 𝑋) = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
22	20	oveqi 7412	. . 3 ⊢ (⟨𝑌, 𝑋⟩ · 𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
23	12, 13, 14, 15, 16, 17, 18, 19, 21, 22	dfiso2 17770	. 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 1539   ∈ wcel 2107  ∃wrex 3059  ⟨cop 4605  ‘cfv 6527  (class class class)co 7399  Basecbs 17213  Hom chom 17267  compcco 17268  Catccat 17661  Idccid 17662  Isociso 17744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5246  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-iun 4966  df-br 5117  df-opab 5179  df-mpt 5199  df-id 5545  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 6480  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7402  df-oprab 7403  df-mpo 7404  df-1st 7982  df-2nd 7983  df-sect 17745  df-inv 17746  df-iso 17747

This theorem is referenced by:  upciclem4  48925