Theorem dfiso2 16871

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 2795	. . . 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 16864	. . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌))
8	7	eleq2d 2868	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
9		eqid 2795	. . . . 5 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
10	1, 2, 3, 4, 5, 9	invfval 16858	. . . 4 ⊢ (𝜑 → (𝑋(Inv‘𝐶)𝑌) = ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋)))
11	10	dmeqd 5660	. . 3 ⊢ (𝜑 → dom (𝑋(Inv‘𝐶)𝑌) = dom ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋)))
12	11	eleq2d 2868	. 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 2795	. . . . . . . . 9 ⊢ (comp‘𝐶) = (comp‘𝐶)
15		dfiso2.1	. . . . . . . . 9 ⊢ 1 = (Id‘𝐶)
16	1, 13, 14, 15, 9, 3, 4, 5	sectfval 16850	. . . . . . . 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 16850	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌(Sect‘𝐶)𝑋) = {⟨𝑔, 𝑓⟩ ∣ ((𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌)) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))})
18	17	cnveqd 5632	. . . . . . . . 9 ⊢ (𝜑 → ◡(𝑌(Sect‘𝐶)𝑋) = ◡{⟨𝑔, 𝑓⟩ ∣ ((𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌)) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))})
19		cnvopab 5873	. . . . . . . . 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 2847	. . . . . . . 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 4110	. . . . . . 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 5587	. . . . . . . 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 652	. . . . . . . . . 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 653	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌))) ↔ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))))
25		anidm 565	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))) ↔ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)))
26	24, 25	bitri 276	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌))) ↔ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)))
27	26	anbi1i 623	. . . . . . . . . 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 276	. . . . . . . . 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 5029	. . . . . . . 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 2819	. . . . . . 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 2847	. . . . . 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 5660	. . . . 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 5670	. . . . 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 2847	. . . 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 2868	. . 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 2870	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ (𝑋𝐻𝑌) ↔ 𝐹 ∈ (𝑋𝐻𝑌)))
38	37	anbi1d 629	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))))
39		oveq2 7024	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹))
40	39	eqeq1d 2797	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ↔ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋)))
41		oveq1 7023	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔))
42	41	eqeq1d 2797	. . . . . . . 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 630	. . . . . . 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 630	. . . . . 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 1899	. . . . 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 3604	. . . 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 533	. . . . . . 7 ⊢ (𝜑 → (𝑔 ∈ (𝑌𝐻𝑋) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))))
49	48	bicomd 224	. . . . . 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 2801	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋) = ⚬ )
53	52	oveqd 7033	. . . . . . . 8 ⊢ (𝜑 → (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = (𝑔 ⚬ 𝐹))
54	53	eqeq1d 2797	. . . . . . 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 2801	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = ∗ )
58	57	oveqd 7033	. . . . . . . 8 ⊢ (𝜑 → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹 ∗ 𝑔))
59	58	eqeq1d 2797	. . . . . . 7 ⊢ (𝜑 → ((𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌) ↔ (𝐹 ∗ 𝑔) = ( 1 ‘𝑌)))
60	54, 59	anbi12d 630	. . . . . 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 630	. . . . 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 1899	. . . 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 3111	. . . 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 290	. . 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 306	. 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 306	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)((𝑔 ⚬ 𝐹) = ( 1 ‘𝑋) ∧ (𝐹 ∗ 𝑔) = ( 1 ‘𝑌))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522  ∃wex 1761   ∈ wcel 2081  {cab 2775  ∃wrex 3106   ∩ cin 3858  ⟨cop 4478  {copab 5024  ^◡ccnv 5442  dom cdm 5443  ‘cfv 6225  (class class class)co 7016  Basecbs 16312  Hom chom 16405  compcco 16406  Catccat 16764  Idccid 16765  Sectcsect 16843  Invcinv 16844  Isociso 16845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-1st 7545  df-2nd 7546  df-sect 16846  df-inv 16847  df-iso 16848

This theorem is referenced by:  dfiso3  16872