Theorem dfiso2 16432

Description: Alternate definition of an isomorphism of a category, according to definition 3.8 in [Adamek] p. 28. (Contributed by AV, 10-Apr-2017.)

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 2622	. . . 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 16425	. . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌))
8	7	eleq2d 2687	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
9		eqid 2622	. . . . 5 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
10	1, 2, 3, 4, 5, 9	invfval 16419	. . . 4 ⊢ (𝜑 → (𝑋(Inv‘𝐶)𝑌) = ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋)))
11	10	dmeqd 5326	. . 3 ⊢ (𝜑 → dom (𝑋(Inv‘𝐶)𝑌) = dom ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋)))
12	11	eleq2d 2687	. 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 2622	. . . . . . . . 9 ⊢ (comp‘𝐶) = (comp‘𝐶)
15		dfiso2.1	. . . . . . . . 9 ⊢ 1 = (Id‘𝐶)
16	1, 13, 14, 15, 9, 3, 4, 5	sectfval 16411	. . . . . . . 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 16411	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌(Sect‘𝐶)𝑋) = {⟨𝑔, 𝑓⟩ ∣ ((𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌)) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))})
18	17	cnveqd 5298	. . . . . . . . 9 ⊢ (𝜑 → ◡(𝑌(Sect‘𝐶)𝑋) = ◡{⟨𝑔, 𝑓⟩ ∣ ((𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌)) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))})
19		cnvopab 5533	. . . . . . . . 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	syl6eq 2672	. . . . . . . 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 3815	. . . . . . 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 5252	. . . . . . . 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 865	. . . . . . . . . 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 866	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌))) ↔ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))))
25		anidm 676	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))) ↔ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)))
26	24, 25	bitri 264	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌))) ↔ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)))
27	26	anbi1i 731	. . . . . . . . . 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 264	. . . . . . . . 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 4717	. . . . . . . 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 2644	. . . . . . 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	syl6eq 2672	. . . . . 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 5326	. . . . 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 5335	. . . . 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	syl6eq 2672	. . . 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 2687	. . 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 2689	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ (𝑋𝐻𝑌) ↔ 𝐹 ∈ (𝑋𝐻𝑌)))
38	37	anbi1d 741	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))))
39		oveq2 6658	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹))
40	39	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ↔ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋)))
41		oveq1 6657	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔))
42	41	eqeq1d 2624	. . . . . . . 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 747	. . . . . . 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 747	. . . . . 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 1850	. . . . 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 3351	. . . 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 17	. . 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 529	. . . . . . 7 ⊢ (𝜑 → (𝑔 ∈ (𝑌𝐻𝑋) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))))
49	48	bicomd 213	. . . . . 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 2628	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋) = ⚬ )
53	52	oveqd 6667	. . . . . . . 8 ⊢ (𝜑 → (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = (𝑔 ⚬ 𝐹))
54	53	eqeq1d 2624	. . . . . . 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 2628	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = ∗ )
58	57	oveqd 6667	. . . . . . . 8 ⊢ (𝜑 → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹 ∗ 𝑔))
59	58	eqeq1d 2624	. . . . . . 7 ⊢ (𝜑 → ((𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌) ↔ (𝐹 ∗ 𝑔) = ( 1 ‘𝑌)))
60	54, 59	anbi12d 747	. . . . . 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 747	. . . . 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 1850	. . . 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 2918	. . . 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	syl6bbr 278	. . 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 294	. 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 294	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)((𝑔 ⚬ 𝐹) = ( 1 ‘𝑋) ∧ (𝐹 ∗ 𝑔) = ( 1 ‘𝑌))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∃wrex 2913   ∩ cin 3573  ⟨cop 4183  {copab 4712  ^◡ccnv 5113  dom cdm 5114  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  Hom chom 15952  compcco 15953  Catccat 16325  Idccid 16326  Sectcsect 16404  Invcinv 16405  Isociso 16406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-sect 16407  df-inv 16408  df-iso 16409

This theorem is referenced by:  dfiso3  16433