Theorem isisod 49386

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 7383	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹))
6	5	eqeq1d 2739	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → ((𝑔(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋) ↔ (𝐺(⟨𝑋, 𝑌⟩ · 𝑋)𝐹) = ( 1 ‘𝑋)))
7	4	oveq2d 7384	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝑔) = (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝐺))
8	7	eqeq1d 2739	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → ((𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝑔) = ( 1 ‘𝑌) ↔ (𝐹(⟨𝑌, 𝑋⟩ · 𝑌)𝐺) = ( 1 ‘𝑌)))
9	6, 8	anbi12d 633	. . . 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 3571	. . 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 700	. 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 7381	. . 3 ⊢ (⟨𝑋, 𝑌⟩ · 𝑋) = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
22	20	oveqi 7381	. . 3 ⊢ (⟨𝑌, 𝑋⟩ · 𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
23	12, 13, 14, 15, 16, 17, 18, 19, 21, 22	dfiso2 17708	. 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 1542   ∈ wcel 2114  ∃wrex 3062  ⟨cop 4588  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  Hom chom 17200  compcco 17201  Catccat 17599  Idccid 17600  Isociso 17682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-sect 17683  df-inv 17684  df-iso 17685

This theorem is referenced by:  upciclem4  49528