Theorem dfiso2 17758

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 2725	. . . 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 17751	. . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋(Inv‘𝐶)𝑌))
8	7	eleq2d 2811	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ dom (𝑋(Inv‘𝐶)𝑌)))
9		eqid 2725	. . . . 5 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
10	1, 2, 3, 4, 5, 9	invfval 17745	. . . 4 ⊢ (𝜑 → (𝑋(Inv‘𝐶)𝑌) = ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋)))
11	10	dmeqd 5908	. . 3 ⊢ (𝜑 → dom (𝑋(Inv‘𝐶)𝑌) = dom ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋)))
12	11	eleq2d 2811	. 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 2725	. . . . . . . . 9 ⊢ (comp‘𝐶) = (comp‘𝐶)
15		dfiso2.1	. . . . . . . . 9 ⊢ 1 = (Id‘𝐶)
16	1, 13, 14, 15, 9, 3, 4, 5	sectfval 17737	. . . . . . . 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 17737	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌(Sect‘𝐶)𝑋) = {⟨𝑔, 𝑓⟩ ∣ ((𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌)) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))})
18	17	cnveqd 5878	. . . . . . . . 9 ⊢ (𝜑 → ◡(𝑌(Sect‘𝐶)𝑋) = ◡{⟨𝑔, 𝑓⟩ ∣ ((𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌)) ∧ (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = ( 1 ‘𝑌))})
19		cnvopab 6144	. . . . . . . . 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 2781	. . . . . . . 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 4211	. . . . . . 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 5831	. . . . . . . 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 654	. . . . . . . . . 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 655	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌))) ↔ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))))
25		anidm 563	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))) ↔ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)))
26	24, 25	bitri 274	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔 ∈ (𝑌𝐻𝑋) ∧ 𝑓 ∈ (𝑋𝐻𝑌))) ↔ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)))
27	26	anbi1i 622	. . . . . . . . . 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 274	. . . . . . . . 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 5216	. . . . . . . 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 2753	. . . . . . 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 2781	. . . . . 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 5908	. . . . 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 5918	. . . . 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 2781	. . . 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 2811	. . 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 2813	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ (𝑋𝐻𝑌) ↔ 𝐹 ∈ (𝑋𝐻𝑌)))
38	37	anbi1d 629	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))))
39		oveq2 7427	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹))
40	39	eqeq1d 2727	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝑓) = ( 1 ‘𝑋) ↔ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋)))
41		oveq1 7426	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔))
42	41	eqeq1d 2727	. . . . . . . 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 1916	. . . . 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 3662	. . . 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 531	. . . . . . 7 ⊢ (𝜑 → (𝑔 ∈ (𝑌𝐻𝑋) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))))
49	48	bicomd 222	. . . . . 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 2731	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋) = ⚬ )
53	52	oveqd 7436	. . . . . . . 8 ⊢ (𝜑 → (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = (𝑔 ⚬ 𝐹))
54	53	eqeq1d 2727	. . . . . . 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 2731	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = ∗ )
58	57	oveqd 7436	. . . . . . . 8 ⊢ (𝜑 → (𝐹(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝐹 ∗ 𝑔))
59	58	eqeq1d 2727	. . . . . . 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 1916	. . . 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 3060	. . . 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 288	. . 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 304	. 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 304	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)((𝑔 ⚬ 𝐹) = ( 1 ‘𝑋) ∧ (𝐹 ∗ 𝑔) = ( 1 ‘𝑌))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2702  ∃wrex 3059   ∩ cin 3943  ⟨cop 4636  {copab 5211  ^◡ccnv 5677  dom cdm 5678  ‘cfv 6549  (class class class)co 7419  Basecbs 17183  Hom chom 17247  compcco 17248  Catccat 17647  Idccid 17648  Sectcsect 17730  Invcinv 17731  Isociso 17732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-sect 17733  df-inv 17734  df-iso 17735

This theorem is referenced by:  dfiso3  17759