Theorem isisod 49805

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 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
5	4	oveq1d 7425	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹))
6	5	eqeq1d 2765	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → ((𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋) ↔ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋)))
7	4	oveq2d 7426	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝑔) = (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝐺))
8	7	eqeq1d 2765	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → ((𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝑔) = ( 1 ‘𝑌) ↔ (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝐺) = ( 1 ‘𝑌)))
9	6, 8	anbi12d 643	. . . 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 3574	. . 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 711	. 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 7423	. . 3 ⊢ (⟨𝑋, 𝑌⟩ · 𝑋) = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
22	20	oveqi 7423	. . 3 ⊢ (⟨𝑌, 𝑋⟩ · 𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
23	12, 13, 14, 15, 16, 17, 18, 19, 21, 22	dfiso2 17824	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)((𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝑔) = ( 1 ‘𝑌))))
24	11, 23	mpbird 260	1 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  ⟨cop 4595  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  Hom chom 17316  compcco 17317  Catccat 17715  Idccid 17716  Isociso 17798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-sect 17799  df-inv 17800  df-iso 17801

This theorem is referenced by:  upciclem4  49947