Theorem dfiso2 17830

Description: Alternate definition of an isomorphism of a category, according to definition 3.8 in [Adamek] p. 28. (Contributed by AV, 10-Apr-2020.)

Hypotheses

Ref	Expression
dfiso2.b	⊢ 𝐵 = (Base‘𝐶)
dfiso2.h	⊢ 𝐻 = (Hom ‘𝐶)
dfiso2.c	⊢ (𝜑 → 𝐶 ∈ Cat)
dfiso2.i	⊢ 𝐼 = (Iso‘𝐶)
dfiso2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
dfiso2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
dfiso2.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
dfiso2.1	⊢ 1 = (Id‘𝐶)
dfiso2.o	⊢ ⚬ = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
dfiso2.p	⊢ ∗ = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)

Assertion

Ref	Expression
dfiso2	⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)((𝑔 ⚬ 𝐹) = ( 1 ‘𝑋) ∧ (𝐹 ∗ 𝑔) = ( 1 ‘𝑌))))

Distinct variable groups:   𝐶,𝑔   𝑔,𝐹   𝑔,𝐻   𝑔,𝐼   𝑔,𝑋   𝑔,𝑌   ⚬ ,𝑔   ∗ ,𝑔   1 ,𝑔   𝜑,𝑔

Allowed substitution hint:   𝐵(𝑔)

Proof of Theorem dfiso2

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiso2.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		eqid 2763	. . . 4 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
3		dfiso2.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		dfiso2.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		dfiso2.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6		dfiso2.i	. . . 4 ⊢ 𝐼 = (Iso‘𝐶)
7	1, 2, 3, 4, 5, 6	isoval 17823	. . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌))
8	7	eleq2d 2849	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
9		eqid 2763	. . . . 5 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
10	1, 2, 3, 4, 5, 9	invfval 17817	. . . 4 ⊢ (𝜑 → (𝑋(Inv‘𝐶)𝑌) = ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋)))
11	10	dmeqd 5897	. . 3 ⊢ (𝜑 → dom (𝑋(Inv‘𝐶)𝑌) = dom ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋)))
12	11	eleq2d 2849	. 2 ⊢ (𝜑 → (𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌) ↔ 𝐹 ∈ dom ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋))))
13		dfiso2.h	. . . . . . . . 9 ⊢ 𝐻 = (Hom ‘𝐶)
14		eqid 2763	. . . . . . . . 9 ⊢ (comp‘𝐶) = (comp‘𝐶)
15		dfiso2.1	. . . . . . . . 9 ⊢ 1 = (Id‘𝐶)
16	1, 13, 14, 15, 9, 3, 4, 5	sectfval 17809	. . . . . . . 8 ⊢ (𝜑 → (𝑋(Sect‘𝐶)𝑌) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋))})
17	1, 13, 14, 15, 9, 3, 5, 4	sectfval 17809	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌(Sect‘𝐶)𝑋) = {⟨𝑔, 𝑓⟩ ∣ ((𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌)) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))})
18	17	cnveqd 5863	. . . . . . . . 9 ⊢ (𝜑 → ◡(𝑌(Sect‘𝐶)𝑋) = ◡{⟨𝑔, 𝑓⟩ ∣ ((𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌)) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))})
19		cnvopab 6139	. . . . . . . . 9 ⊢ ◡{⟨𝑔, 𝑓⟩ ∣ ((𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌)) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌)) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))}
20	18, 19	eqtrdi 2814	. . . . . . . 8 ⊢ (𝜑 → ◡(𝑌(Sect‘𝐶)𝑋) = {⟨𝑓, 𝑔⟩ ∣ ((𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌)) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))})
21	16, 20	ineq12d 4175	. . . . . . 7 ⊢ (𝜑 → ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋)) = ({⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋))} ∩ {⟨𝑓, 𝑔⟩ ∣ ((𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌)) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))}))
22		inopab 5818	. . . . . . . 8 ⊢ ({⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋))} ∩ {⟨𝑓, 𝑔⟩ ∣ ((𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌)) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))}) = {⟨𝑓, 𝑔⟩ ∣ (((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋)) ∧ ((𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌)) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌)))}
23		an4 668	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋)) ∧ ((𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌)) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))) ↔ (((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌))) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))))
24		an42 669	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌))) ↔ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))))
25		anidm 574	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))) ↔ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)))
26	24, 25	bitri 278	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌))) ↔ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)))
27	26	anbi1i 635	. . . . . . . . . 10 ⊢ ((((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌))) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))) ↔ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))))
28	23, 27	bitri 278	. . . . . . . . 9 ⊢ ((((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋)) ∧ ((𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌)) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))) ↔ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))))
29	28	opabbii 5179	. . . . . . . 8 ⊢ {⟨𝑓, 𝑔⟩ ∣ (((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋)) ∧ ((𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌)) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌)))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌)))}
30	22, 29	eqtri 2786	. . . . . . 7 ⊢ ({⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋))} ∩ {⟨𝑓, 𝑔⟩ ∣ ((𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌)) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))}) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌)))}
31	21, 30	eqtrdi 2814	. . . . . 6 ⊢ (𝜑 → ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋)) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌)))})
32	31	dmeqd 5897	. . . . 5 ⊢ (𝜑 → dom ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋)) = dom {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌)))})
33		dmopab 5907	. . . . 5 ⊢ dom {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌)))} = {𝑓 ∣ ∃𝑔((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌)))}
34	32, 33	eqtrdi 2814	. . . 4 ⊢ (𝜑 → dom ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋)) = {𝑓 ∣ ∃𝑔((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌)))})
35	34	eleq2d 2849	. . 3 ⊢ (𝜑 → (𝐹 ∈ dom ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋)) ↔ 𝐹 ∈ {𝑓 ∣ ∃𝑔((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌)))}))
36		dfiso2.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
37		eleq1 2851	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ (𝑋𝐻𝑌) ↔ 𝐹 ∈ (𝑋𝐻𝑌)))
38	37	anbi1d 642	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))))
39		oveq2 7420	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹))
40	39	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ↔ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋)))
41		oveq1 7419	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔))
42	41	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌) ↔ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌)))
43	40, 42	anbi12d 643	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌)) ↔ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))))
44	38, 43	anbi12d 643	. . . . . 6 ⊢ (𝑓 = 𝐹 → (((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))) ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌)))))
45	44	exbidv 1951	. . . . 5 ⊢ (𝑓 = 𝐹 → (∃𝑔((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))) ↔ ∃𝑔((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌)))))
46	45	elabg 3636	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝐹 ∈ {𝑓 ∣ ∃𝑔((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌)))} ↔ ∃𝑔((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌)))))
47	36, 46	syl 18	. . 3 ⊢ (𝜑 → (𝐹 ∈ {𝑓 ∣ ∃𝑔((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌)))} ↔ ∃𝑔((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌)))))
48	36	biantrurd 541	. . . . . . 7 ⊢ (𝜑 → (𝑔 ∈ (𝑌𝐻𝑋) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))))
49	48	bicomd 226	. . . . . 6 ⊢ (𝜑 → ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ↔ 𝑔 ∈ (𝑌𝐻𝑋)))
50		dfiso2.o	. . . . . . . . . . 11 ⊢ ⚬ = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)
51	50	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ⚬ = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋))
52	51	eqcomd 2769	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋) = ⚬ )
53	52	oveqd 7429	. . . . . . . 8 ⊢ (𝜑 → (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = (𝑔 ⚬ 𝐹))
54	53	eqeq1d 2765	. . . . . . 7 ⊢ (𝜑 → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋) ↔ (𝑔 ⚬ 𝐹) = ( 1 ‘𝑋)))
55		dfiso2.p	. . . . . . . . . . 11 ⊢ ∗ = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
56	55	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ∗ = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌))
57	56	eqcomd 2769	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = ∗ )
58	57	oveqd 7429	. . . . . . . 8 ⊢ (𝜑 → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹 ∗ 𝑔))
59	58	eqeq1d 2765	. . . . . . 7 ⊢ (𝜑 → ((𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌) ↔ (𝐹 ∗ 𝑔) = ( 1 ‘𝑌)))
60	54, 59	anbi12d 643	. . . . . 6 ⊢ (𝜑 → (((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌)) ↔ ((𝑔 ⚬ 𝐹) = ( 1 ‘𝑋) ∧ (𝐹 ∗ 𝑔) = ( 1 ‘𝑌))))
61	49, 60	anbi12d 643	. . . . 5 ⊢ (𝜑 → (((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))) ↔ (𝑔 ∈ (𝑌𝐻𝑋) ∧ ((𝑔 ⚬ 𝐹) = ( 1 ‘𝑋) ∧ (𝐹 ∗ 𝑔) = ( 1 ‘𝑌)))))
62	61	exbidv 1951	. . . 4 ⊢ (𝜑 → (∃𝑔((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))) ↔ ∃𝑔(𝑔 ∈ (𝑌𝐻𝑋) ∧ ((𝑔 ⚬ 𝐹) = ( 1 ‘𝑋) ∧ (𝐹 ∗ 𝑔) = ( 1 ‘𝑌)))))
63		df-rex 3090	. . . 4 ⊢ (∃𝑔 ∈ (𝑌𝐻𝑋)((𝑔 ⚬ 𝐹) = ( 1 ‘𝑋) ∧ (𝐹 ∗ 𝑔) = ( 1 ‘𝑌)) ↔ ∃𝑔(𝑔 ∈ (𝑌𝐻𝑋) ∧ ((𝑔 ⚬ 𝐹) = ( 1 ‘𝑋) ∧ (𝐹 ∗ 𝑔) = ( 1 ‘𝑌))))
64	62, 63	bitr4di 292	. . 3 ⊢ (𝜑 → (∃𝑔((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋) ∧ (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)((𝑔 ⚬ 𝐹) = ( 1 ‘𝑋) ∧ (𝐹 ∗ 𝑔) = ( 1 ‘𝑌))))
65	35, 47, 64	3bitrd 308	. 2 ⊢ (𝜑 → (𝐹 ∈ dom ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋)) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)((𝑔 ⚬ 𝐹) = ( 1 ‘𝑋) ∧ (𝐹 ∗ 𝑔) = ( 1 ‘𝑌))))
66	8, 12, 65	3bitrd 308	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)((𝑔 ⚬ 𝐹) = ( 1 ‘𝑋) ∧ (𝐹 ∗ 𝑔) = ( 1 ‘𝑌))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  ∃wrex 3089   ∩ cin 3905  ⟨cop 4596  {copab 5174  ^◡ccnv 5662  dom cdm 5663  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  Hom chom 17322  compcco 17323  Catccat 17721  Idccid 17722  Sectcsect 17802  Invcinv 17803  Isociso 17804

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 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

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 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-sect 17805  df-inv 17806  df-iso 17807

This theorem is referenced by:  dfiso3  17831  isisod  49788