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Theorem ptpjopn 23560

Description: The projection map is an open map. (Contributed by Mario Carneiro, 2-Sep-2015.)

**Hypotheses**
Ref	Expression
ptpjcn.1	⊢ 𝑌 = ∪ 𝐽
ptpjcn.2	⊢ 𝐽 = (∏_t‘𝐹)

**Assertion**
Ref	Expression
ptpjopn	⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) → ((𝑥 ∈ 𝑌 ↦ (𝑥‘𝐼)) “ 𝑈) ∈ (𝐹‘𝐼))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐹 𝑥,𝐼 𝑥,𝑉 𝑥,𝑌 𝑥,𝑈 Allowed substitution hint: 𝐽(𝑥)

Proof of Theorem ptpjopn Dummy variables 𝑔 𝑘 𝑛 𝑠 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ima 5691	. . 3 ⊢ ((𝑥 ∈ 𝑌 ↦ (𝑥‘𝐼)) “ 𝑈) = ran ((𝑥 ∈ 𝑌 ↦ (𝑥‘𝐼)) ↾ 𝑈)
2		elssuni 4941	. . . . . . 7 ⊢ (𝑈 ∈ 𝐽 → 𝑈 ⊆ ∪ 𝐽)
3		ptpjcn.1	. . . . . . 7 ⊢ 𝑌 = ∪ 𝐽
4	2, 3	sseqtrrdi 4028	. . . . . 6 ⊢ (𝑈 ∈ 𝐽 → 𝑈 ⊆ 𝑌)
5	4	adantl 480	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) → 𝑈 ⊆ 𝑌)
6	5	resmptd 6045	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) → ((𝑥 ∈ 𝑌 ↦ (𝑥‘𝐼)) ↾ 𝑈) = (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)))
7	6	rneqd 5940	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) → ran ((𝑥 ∈ 𝑌 ↦ (𝑥‘𝐼)) ↾ 𝑈) = ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)))
8	1, 7	eqtrid 2777	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) → ((𝑥 ∈ 𝑌 ↦ (𝑥‘𝐼)) “ 𝑈) = ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)))
9		ptpjcn.2	. . . . . . . . . . 11 ⊢ 𝐽 = (∏_t‘𝐹)
10		ffn 6723	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶Top → 𝐹 Fn 𝐴)
11		eqid 2725	. . . . . . . . . . . . 13 ⊢ {𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑠 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} = {𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑠 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}
12	11	ptval 23518	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) → (∏_t‘𝐹) = (topGen‘{𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑠 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
13	10, 12	sylan2 591	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∏_t‘𝐹) = (topGen‘{𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑠 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
14	9, 13	eqtrid 2777	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐽 = (topGen‘{𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑠 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
15	14	3adant3 1129	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → 𝐽 = (topGen‘{𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑠 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
16	15	eleq2d 2811	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) → (𝑈 ∈ 𝐽 ↔ 𝑈 ∈ (topGen‘{𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑠 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))})))
17	16	biimpa 475	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) → 𝑈 ∈ (topGen‘{𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑠 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
18		tg2 22912	. . . . . . 7 ⊢ ((𝑈 ∈ (topGen‘{𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑠 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}) ∧ 𝑠 ∈ 𝑈) → ∃𝑤 ∈ {𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑠 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} (𝑠 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑈))
19	17, 18	sylan 578	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) → ∃𝑤 ∈ {𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑠 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} (𝑠 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑈))
20		vex 3465	. . . . . . . . 9 ⊢ 𝑤 ∈ V
21		eqeq1 2729	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑤 → (𝑠 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) ↔ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)))
22	21	anbi2d 628	. . . . . . . . . 10 ⊢ (𝑠 = 𝑤 → (((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑠 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) ↔ ((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))))
23	22	exbidv 1916	. . . . . . . . 9 ⊢ (𝑠 = 𝑤 → (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑠 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) ↔ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))))
24	20, 23	elab 3664	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑠 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} ↔ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)))
25		fveq2 6896	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐼 → (𝑔‘𝑦) = (𝑔‘𝐼))
26		fveq2 6896	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐼 → (𝐹‘𝑦) = (𝐹‘𝐼))
27	25, 26	eleq12d 2819	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐼 → ((𝑔‘𝑦) ∈ (𝐹‘𝑦) ↔ (𝑔‘𝐼) ∈ (𝐹‘𝐼)))
28		simplr2 1213	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) → ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦))
29		simpl3 1190	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) → 𝐼 ∈ 𝐴)
30	29	ad3antrrr 728	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) → 𝐼 ∈ 𝐴)
31	27, 28, 30	rspcdva 3607	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) → (𝑔‘𝐼) ∈ (𝐹‘𝐼))
32		fveq2 6896	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐼 → (𝑠‘𝑦) = (𝑠‘𝐼))
33	32, 25	eleq12d 2819	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐼 → ((𝑠‘𝑦) ∈ (𝑔‘𝑦) ↔ (𝑠‘𝐼) ∈ (𝑔‘𝐼)))
34		vex 3465	. . . . . . . . . . . . . . . . 17 ⊢ 𝑠 ∈ V
35	34	elixp 8923	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ↔ (𝑠 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑠‘𝑦) ∈ (𝑔‘𝑦)))
36	35	simprbi 495	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) → ∀𝑦 ∈ 𝐴 (𝑠‘𝑦) ∈ (𝑔‘𝑦))
37	36	ad2antrl 726	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) → ∀𝑦 ∈ 𝐴 (𝑠‘𝑦) ∈ (𝑔‘𝑦))
38	33, 37, 30	rspcdva 3607	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) → (𝑠‘𝐼) ∈ (𝑔‘𝐼))
39		simplrr 776	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔‘𝐼)) → X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)
40		simplrl 775	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ (𝑘 ∈ (𝑔‘𝐼) ∧ 𝑛 ∈ 𝐴)) ∧ 𝑛 = 𝐼) → 𝑘 ∈ (𝑔‘𝐼))
41		fveq2 6896	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝐼 → (𝑔‘𝑛) = (𝑔‘𝐼))
42	41	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ (𝑘 ∈ (𝑔‘𝐼) ∧ 𝑛 ∈ 𝐴)) ∧ 𝑛 = 𝐼) → (𝑔‘𝑛) = (𝑔‘𝐼))
43	40, 42	eleqtrrd 2828	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ (𝑘 ∈ (𝑔‘𝐼) ∧ 𝑛 ∈ 𝐴)) ∧ 𝑛 = 𝐼) → 𝑘 ∈ (𝑔‘𝑛))
44		fveq2 6896	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = 𝑛 → (𝑠‘𝑦) = (𝑠‘𝑛))
45		fveq2 6896	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = 𝑛 → (𝑔‘𝑦) = (𝑔‘𝑛))
46	44, 45	eleq12d 2819	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑛 → ((𝑠‘𝑦) ∈ (𝑔‘𝑦) ↔ (𝑠‘𝑛) ∈ (𝑔‘𝑛)))
47		simplrl 775	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ (𝑘 ∈ (𝑔‘𝐼) ∧ 𝑛 ∈ 𝐴)) → 𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))
48	47, 36	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ (𝑘 ∈ (𝑔‘𝐼) ∧ 𝑛 ∈ 𝐴)) → ∀𝑦 ∈ 𝐴 (𝑠‘𝑦) ∈ (𝑔‘𝑦))
49		simprr 771	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ (𝑘 ∈ (𝑔‘𝐼) ∧ 𝑛 ∈ 𝐴)) → 𝑛 ∈ 𝐴)
50	46, 48, 49	rspcdva 3607	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ (𝑘 ∈ (𝑔‘𝐼) ∧ 𝑛 ∈ 𝐴)) → (𝑠‘𝑛) ∈ (𝑔‘𝑛))
51	50	adantr 479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ (𝑘 ∈ (𝑔‘𝐼) ∧ 𝑛 ∈ 𝐴)) ∧ ¬ 𝑛 = 𝐼) → (𝑠‘𝑛) ∈ (𝑔‘𝑛))
52	43, 51	ifclda 4565	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ (𝑘 ∈ (𝑔‘𝐼) ∧ 𝑛 ∈ 𝐴)) → if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛)) ∈ (𝑔‘𝑛))
53	52	anassrs 466	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔‘𝐼)) ∧ 𝑛 ∈ 𝐴) → if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛)) ∈ (𝑔‘𝑛))
54	53	ralrimiva 3135	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔‘𝐼)) → ∀𝑛 ∈ 𝐴 if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛)) ∈ (𝑔‘𝑛))
55		simpll1 1209	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) → 𝐴 ∈ 𝑉)
56	55	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔‘𝐼)) → 𝐴 ∈ 𝑉)
57		mptelixpg 8954	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ 𝑉 → ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛))) ∈ X𝑛 ∈ 𝐴 (𝑔‘𝑛) ↔ ∀𝑛 ∈ 𝐴 if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛)) ∈ (𝑔‘𝑛)))
58	56, 57	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔‘𝐼)) → ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛))) ∈ X𝑛 ∈ 𝐴 (𝑔‘𝑛) ↔ ∀𝑛 ∈ 𝐴 if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛)) ∈ (𝑔‘𝑛)))
59	54, 58	mpbird 256	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔‘𝐼)) → (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛))) ∈ X𝑛 ∈ 𝐴 (𝑔‘𝑛))
60		fveq2 6896	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑦 → (𝑔‘𝑛) = (𝑔‘𝑦))
61	60	cbvixpv 8934	. . . . . . . . . . . . . . . . . . 19 ⊢ X𝑛 ∈ 𝐴 (𝑔‘𝑛) = X𝑦 ∈ 𝐴 (𝑔‘𝑦)
62	59, 61	eleqtrdi 2835	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔‘𝐼)) → (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛))) ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦))
63	39, 62	sseldd 3977	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔‘𝐼)) → (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛))) ∈ 𝑈)
64	30	adantr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔‘𝐼)) → 𝐼 ∈ 𝐴)
65		iftrue 4536	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝐼 → if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛)) = 𝑘)
66		eqid 2725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛))) = (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛)))
67		vex 3465	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑘 ∈ V
68	65, 66, 67	fvmpt 7004	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ 𝐴 → ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛)))‘𝐼) = 𝑘)
69	64, 68	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔‘𝐼)) → ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛)))‘𝐼) = 𝑘)
70	69	eqcomd 2731	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔‘𝐼)) → 𝑘 = ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛)))‘𝐼))
71		fveq1 6895	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛))) → (𝑥‘𝐼) = ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛)))‘𝐼))
72	71	rspceeqv 3628	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛))) ∈ 𝑈 ∧ 𝑘 = ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝐼, 𝑘, (𝑠‘𝑛)))‘𝐼)) → ∃𝑥 ∈ 𝑈 𝑘 = (𝑥‘𝐼))
73	63, 70, 72	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔‘𝐼)) → ∃𝑥 ∈ 𝑈 𝑘 = (𝑥‘𝐼))
74		eqid 2725	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)) = (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))
75	74	elrnmpt 5958	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ V → (𝑘 ∈ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)) ↔ ∃𝑥 ∈ 𝑈 𝑘 = (𝑥‘𝐼)))
76	75	elv 3467	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)) ↔ ∃𝑥 ∈ 𝑈 𝑘 = (𝑥‘𝐼))
77	73, 76	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) ∧ 𝑘 ∈ (𝑔‘𝐼)) → 𝑘 ∈ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)))
78	77	ex 411	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) → (𝑘 ∈ (𝑔‘𝐼) → 𝑘 ∈ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))))
79	78	ssrdv 3982	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) → (𝑔‘𝐼) ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)))
80		eleq2 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑔‘𝐼) → ((𝑠‘𝐼) ∈ 𝑧 ↔ (𝑠‘𝐼) ∈ (𝑔‘𝐼)))
81		sseq1 4002	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑔‘𝐼) → (𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)) ↔ (𝑔‘𝐼) ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))))
82	80, 81	anbi12d 630	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑔‘𝐼) → (((𝑠‘𝐼) ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))) ↔ ((𝑠‘𝐼) ∈ (𝑔‘𝐼) ∧ (𝑔‘𝐼) ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)))))
83	82	rspcev 3606	. . . . . . . . . . . . 13 ⊢ (((𝑔‘𝐼) ∈ (𝐹‘𝐼) ∧ ((𝑠‘𝐼) ∈ (𝑔‘𝐼) ∧ (𝑔‘𝐼) ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)))) → ∃𝑧 ∈ (𝐹‘𝐼)((𝑠‘𝐼) ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))))
84	31, 38, 79, 83	syl12anc 835	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) ∧ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)) → ∃𝑧 ∈ (𝐹‘𝐼)((𝑠‘𝐼) ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))))
85	84	ex 411	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) → ((𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈) → ∃𝑧 ∈ (𝐹‘𝐼)((𝑠‘𝐼) ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)))))
86		eleq2 2814	. . . . . . . . . . . . 13 ⊢ (𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (𝑠 ∈ 𝑤 ↔ 𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦)))
87		sseq1 4002	. . . . . . . . . . . . 13 ⊢ (𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (𝑤 ⊆ 𝑈 ↔ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈))
88	86, 87	anbi12d 630	. . . . . . . . . . . 12 ⊢ (𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → ((𝑠 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑈) ↔ (𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈)))
89	88	imbi1d 340	. . . . . . . . . . 11 ⊢ (𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → (((𝑠 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑈) → ∃𝑧 ∈ (𝐹‘𝐼)((𝑠‘𝐼) ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)))) ↔ ((𝑠 ∈ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ∧ X𝑦 ∈ 𝐴 (𝑔‘𝑦) ⊆ 𝑈) → ∃𝑧 ∈ (𝐹‘𝐼)((𝑠‘𝐼) ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))))))
90	85, 89	syl5ibrcom 246	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))) → (𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦) → ((𝑠 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑈) → ∃𝑧 ∈ (𝐹‘𝐼)((𝑠‘𝐼) ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))))))
91	90	expimpd 452	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) → (((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) → ((𝑠 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑈) → ∃𝑧 ∈ (𝐹‘𝐼)((𝑠‘𝐼) ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))))))
92	91	exlimdv 1928	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) → (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑤 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)) → ((𝑠 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑈) → ∃𝑧 ∈ (𝐹‘𝐼)((𝑠‘𝐼) ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))))))
93	24, 92	biimtrid 241	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) → (𝑤 ∈ {𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑠 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} → ((𝑠 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑈) → ∃𝑧 ∈ (𝐹‘𝐼)((𝑠‘𝐼) ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))))))
94	93	rexlimdv 3142	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) → (∃𝑤 ∈ {𝑠 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑠 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} (𝑠 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑈) → ∃𝑧 ∈ (𝐹‘𝐼)((𝑠‘𝐼) ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)))))
95	19, 94	mpd 15	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) ∧ 𝑠 ∈ 𝑈) → ∃𝑧 ∈ (𝐹‘𝐼)((𝑠‘𝐼) ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))))
96	95	ralrimiva 3135	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) → ∀𝑠 ∈ 𝑈 ∃𝑧 ∈ (𝐹‘𝐼)((𝑠‘𝐼) ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))))
97		fvex 6909	. . . . . 6 ⊢ (𝑠‘𝐼) ∈ V
98	97	rgenw 3054	. . . . 5 ⊢ ∀𝑠 ∈ 𝑈 (𝑠‘𝐼) ∈ V
99		fveq1 6895	. . . . . . 7 ⊢ (𝑥 = 𝑠 → (𝑥‘𝐼) = (𝑠‘𝐼))
100	99	cbvmptv 5262	. . . . . 6 ⊢ (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)) = (𝑠 ∈ 𝑈 ↦ (𝑠‘𝐼))
101		eleq1 2813	. . . . . . . 8 ⊢ (𝑦 = (𝑠‘𝐼) → (𝑦 ∈ 𝑧 ↔ (𝑠‘𝐼) ∈ 𝑧))
102	101	anbi1d 629	. . . . . . 7 ⊢ (𝑦 = (𝑠‘𝐼) → ((𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))) ↔ ((𝑠‘𝐼) ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)))))
103	102	rexbidv 3168	. . . . . 6 ⊢ (𝑦 = (𝑠‘𝐼) → (∃𝑧 ∈ (𝐹‘𝐼)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))) ↔ ∃𝑧 ∈ (𝐹‘𝐼)((𝑠‘𝐼) ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)))))
104	100, 103	ralrnmptw 7103	. . . . 5 ⊢ (∀𝑠 ∈ 𝑈 (𝑠‘𝐼) ∈ V → (∀𝑦 ∈ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))∃𝑧 ∈ (𝐹‘𝐼)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))) ↔ ∀𝑠 ∈ 𝑈 ∃𝑧 ∈ (𝐹‘𝐼)((𝑠‘𝐼) ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)))))
105	98, 104	ax-mp 5	. . . 4 ⊢ (∀𝑦 ∈ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))∃𝑧 ∈ (𝐹‘𝐼)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))) ↔ ∀𝑠 ∈ 𝑈 ∃𝑧 ∈ (𝐹‘𝐼)((𝑠‘𝐼) ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))))
106	96, 105	sylibr 233	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) → ∀𝑦 ∈ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))∃𝑧 ∈ (𝐹‘𝐼)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))))
107		simpl2 1189	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) → 𝐹:𝐴⟶Top)
108	107, 29	ffvelcdmd 7094	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) → (𝐹‘𝐼) ∈ Top)
109		eltop2 22922	. . . 4 ⊢ ((𝐹‘𝐼) ∈ Top → (ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)) ∈ (𝐹‘𝐼) ↔ ∀𝑦 ∈ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))∃𝑧 ∈ (𝐹‘𝐼)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)))))
110	108, 109	syl 17	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) → (ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)) ∈ (𝐹‘𝐼) ↔ ∀𝑦 ∈ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼))∃𝑧 ∈ (𝐹‘𝐼)(𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)))))
111	106, 110	mpbird 256	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) → ran (𝑥 ∈ 𝑈 ↦ (𝑥‘𝐼)) ∈ (𝐹‘𝐼))
112	8, 111	eqeltrd 2825	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top ∧ 𝐼 ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) → ((𝑥 ∈ 𝑌 ↦ (𝑥‘𝐼)) “ 𝑈) ∈ (𝐹‘𝐼))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∃wex 1773 ∈ wcel 2098 {cab 2702 ∀wral 3050 ∃wrex 3059 Vcvv 3461 ∖ cdif 3941 ⊆ wss 3944 ifcif 4530 ∪ cuni 4909 ↦ cmpt 5232 ran crn 5679 ↾ cres 5680 “ cima 5681 Fn wfn 6544 ⟶wf 6545 ‘cfv 6549 Xcixp 8916 Fincfn 8964 topGenctg 17422 ∏_tcpt 17423 Topctop 22839

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-ixp 8917 df-topgen 17428 df-pt 17429 df-top 22840

This theorem is referenced by: (None)

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