Theorem ptpcon 30469

Description: The topological product of a collection of path-connected spaces is path-connected. The proof uses the axiom of choice. (Contributed by Mario Carneiro, 17-Feb-2015.)

Assertion

Ref	Expression
ptpcon	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) → (∏t‘𝐹) ∈ PCon)

Proof of Theorem ptpcon

Dummy variables 𝑓 𝑥 𝑦 𝑔 𝑡 𝑧 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pcontop 30461	. . . . 5 ⊢ (𝑥 ∈ PCon → 𝑥 ∈ Top)
2	1	ssriv 3572	. . . 4 ⊢ PCon ⊆ Top
3		fss 5969	. . . 4 ⊢ ((𝐹:𝐴⟶PCon ∧ PCon ⊆ Top) → 𝐹:𝐴⟶Top)
4	2, 3	mpan2 703	. . 3 ⊢ (𝐹:𝐴⟶PCon → 𝐹:𝐴⟶Top)
5		pttop 21195	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∏t‘𝐹) ∈ Top)
6	4, 5	sylan2 490	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) → (∏t‘𝐹) ∈ Top)
7		fvi 6165	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ‘𝐴) = 𝐴)
8	7	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → ( I ‘𝐴) = 𝐴)
9	8	eleq2d 2673	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → (𝑡 ∈ ( I ‘𝐴) ↔ 𝑡 ∈ 𝐴))
10	9	biimpa 500	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ 𝑡 ∈ ( I ‘𝐴)) → 𝑡 ∈ 𝐴)
11		simplr 788	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝐹:𝐴⟶PCon)
12	11	ffvelrnda 6267	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ 𝑡 ∈ 𝐴) → (𝐹‘𝑡) ∈ PCon)
13		simprl 790	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑥 ∈ ∪ (∏t‘𝐹))
14		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (∏t‘𝐹) = (∏t‘𝐹)
15	14	ptuni 21207	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡) = ∪ (∏t‘𝐹))
16	4, 15	sylan2 490	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) → X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡) = ∪ (∏t‘𝐹))
17	16	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡) = ∪ (∏t‘𝐹))
18	13, 17	eleqtrrd 2691	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑥 ∈ X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡))
19		vex 3176	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
20	19	elixp 7801	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡) ↔ (𝑥 Fn 𝐴 ∧ ∀𝑡 ∈ 𝐴 (𝑥‘𝑡) ∈ ∪ (𝐹‘𝑡)))
21	18, 20	sylib 207	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → (𝑥 Fn 𝐴 ∧ ∀𝑡 ∈ 𝐴 (𝑥‘𝑡) ∈ ∪ (𝐹‘𝑡)))
22	21	simprd 478	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → ∀𝑡 ∈ 𝐴 (𝑥‘𝑡) ∈ ∪ (𝐹‘𝑡))
23	22	r19.21bi 2916	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ 𝑡 ∈ 𝐴) → (𝑥‘𝑡) ∈ ∪ (𝐹‘𝑡))
24		simprr 792	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑦 ∈ ∪ (∏t‘𝐹))
25	24, 17	eleqtrrd 2691	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑦 ∈ X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡))
26		vex 3176	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
27	26	elixp 7801	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡) ↔ (𝑦 Fn 𝐴 ∧ ∀𝑡 ∈ 𝐴 (𝑦‘𝑡) ∈ ∪ (𝐹‘𝑡)))
28	25, 27	sylib 207	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → (𝑦 Fn 𝐴 ∧ ∀𝑡 ∈ 𝐴 (𝑦‘𝑡) ∈ ∪ (𝐹‘𝑡)))
29	28	simprd 478	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → ∀𝑡 ∈ 𝐴 (𝑦‘𝑡) ∈ ∪ (𝐹‘𝑡))
30	29	r19.21bi 2916	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ 𝑡 ∈ 𝐴) → (𝑦‘𝑡) ∈ ∪ (𝐹‘𝑡))
31		eqid 2610	. . . . . . . . . 10 ⊢ ∪ (𝐹‘𝑡) = ∪ (𝐹‘𝑡)
32	31	pconcn 30460	. . . . . . . . 9 ⊢ (((𝐹‘𝑡) ∈ PCon ∧ (𝑥‘𝑡) ∈ ∪ (𝐹‘𝑡) ∧ (𝑦‘𝑡) ∈ ∪ (𝐹‘𝑡)) → ∃𝑓 ∈ (II Cn (𝐹‘𝑡))((𝑓‘0) = (𝑥‘𝑡) ∧ (𝑓‘1) = (𝑦‘𝑡)))
33	12, 23, 30, 32	syl3anc 1318	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ 𝑡 ∈ 𝐴) → ∃𝑓 ∈ (II Cn (𝐹‘𝑡))((𝑓‘0) = (𝑥‘𝑡) ∧ (𝑓‘1) = (𝑦‘𝑡)))
34		df-rex 2902	. . . . . . . 8 ⊢ (∃𝑓 ∈ (II Cn (𝐹‘𝑡))((𝑓‘0) = (𝑥‘𝑡) ∧ (𝑓‘1) = (𝑦‘𝑡)) ↔ ∃𝑓(𝑓 ∈ (II Cn (𝐹‘𝑡)) ∧ ((𝑓‘0) = (𝑥‘𝑡) ∧ (𝑓‘1) = (𝑦‘𝑡))))
35	33, 34	sylib 207	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ 𝑡 ∈ 𝐴) → ∃𝑓(𝑓 ∈ (II Cn (𝐹‘𝑡)) ∧ ((𝑓‘0) = (𝑥‘𝑡) ∧ (𝑓‘1) = (𝑦‘𝑡))))
36	10, 35	syldan 486	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ 𝑡 ∈ ( I ‘𝐴)) → ∃𝑓(𝑓 ∈ (II Cn (𝐹‘𝑡)) ∧ ((𝑓‘0) = (𝑥‘𝑡) ∧ (𝑓‘1) = (𝑦‘𝑡))))
37	36	ralrimiva 2949	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → ∀𝑡 ∈ ( I ‘𝐴)∃𝑓(𝑓 ∈ (II Cn (𝐹‘𝑡)) ∧ ((𝑓‘0) = (𝑥‘𝑡) ∧ (𝑓‘1) = (𝑦‘𝑡))))
38		fvex 6113	. . . . . 6 ⊢ ( I ‘𝐴) ∈ V
39		eleq1 2676	. . . . . . 7 ⊢ (𝑓 = (𝑔‘𝑡) → (𝑓 ∈ (II Cn (𝐹‘𝑡)) ↔ (𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡))))
40		fveq1 6102	. . . . . . . . 9 ⊢ (𝑓 = (𝑔‘𝑡) → (𝑓‘0) = ((𝑔‘𝑡)‘0))
41	40	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑓 = (𝑔‘𝑡) → ((𝑓‘0) = (𝑥‘𝑡) ↔ ((𝑔‘𝑡)‘0) = (𝑥‘𝑡)))
42		fveq1 6102	. . . . . . . . 9 ⊢ (𝑓 = (𝑔‘𝑡) → (𝑓‘1) = ((𝑔‘𝑡)‘1))
43	42	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑓 = (𝑔‘𝑡) → ((𝑓‘1) = (𝑦‘𝑡) ↔ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡)))
44	41, 43	anbi12d 743	. . . . . . 7 ⊢ (𝑓 = (𝑔‘𝑡) → (((𝑓‘0) = (𝑥‘𝑡) ∧ (𝑓‘1) = (𝑦‘𝑡)) ↔ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))
45	39, 44	anbi12d 743	. . . . . 6 ⊢ (𝑓 = (𝑔‘𝑡) → ((𝑓 ∈ (II Cn (𝐹‘𝑡)) ∧ ((𝑓‘0) = (𝑥‘𝑡) ∧ (𝑓‘1) = (𝑦‘𝑡))) ↔ ((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡)))))
46	38, 45	ac6s2 9191	. . . . 5 ⊢ (∀𝑡 ∈ ( I ‘𝐴)∃𝑓(𝑓 ∈ (II Cn (𝐹‘𝑡)) ∧ ((𝑓‘0) = (𝑥‘𝑡) ∧ (𝑓‘1) = (𝑦‘𝑡))) → ∃𝑔(𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡)))))
47	37, 46	syl 17	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → ∃𝑔(𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡)))))
48		iitopon 22490	. . . . . . 7 ⊢ II ∈ (TopOn‘(0[,]1))
49	48	a1i 11	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → II ∈ (TopOn‘(0[,]1)))
50		simplll 794	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → 𝐴 ∈ 𝑉)
51	11	adantr 480	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → 𝐹:𝐴⟶PCon)
52	51, 4	syl 17	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → 𝐹:𝐴⟶Top)
53	8	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → ( I ‘𝐴) = 𝐴)
54	53	eleq2d 2673	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → (𝑖 ∈ ( I ‘𝐴) ↔ 𝑖 ∈ 𝐴))
55	54	biimpar 501	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) ∧ 𝑖 ∈ 𝐴) → 𝑖 ∈ ( I ‘𝐴))
56		simprr 792	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))
57		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑖 → (𝑔‘𝑡) = (𝑔‘𝑖))
58		fveq2 6103	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → (𝐹‘𝑡) = (𝐹‘𝑖))
59	58	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑖 → (II Cn (𝐹‘𝑡)) = (II Cn (𝐹‘𝑖)))
60	57, 59	eleq12d 2682	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑖 → ((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ↔ (𝑔‘𝑖) ∈ (II Cn (𝐹‘𝑖))))
61	57	fveq1d 6105	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → ((𝑔‘𝑡)‘0) = ((𝑔‘𝑖)‘0))
62		fveq2 6103	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → (𝑥‘𝑡) = (𝑥‘𝑖))
63	61, 62	eqeq12d 2625	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑖 → (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ↔ ((𝑔‘𝑖)‘0) = (𝑥‘𝑖)))
64	57	fveq1d 6105	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → ((𝑔‘𝑡)‘1) = ((𝑔‘𝑖)‘1))
65		fveq2 6103	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → (𝑦‘𝑡) = (𝑦‘𝑖))
66	64, 65	eqeq12d 2625	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑖 → (((𝑔‘𝑡)‘1) = (𝑦‘𝑡) ↔ ((𝑔‘𝑖)‘1) = (𝑦‘𝑖)))
67	63, 66	anbi12d 743	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑖 → ((((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡)) ↔ (((𝑔‘𝑖)‘0) = (𝑥‘𝑖) ∧ ((𝑔‘𝑖)‘1) = (𝑦‘𝑖))))
68	60, 67	anbi12d 743	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑖 → (((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))) ↔ ((𝑔‘𝑖) ∈ (II Cn (𝐹‘𝑖)) ∧ (((𝑔‘𝑖)‘0) = (𝑥‘𝑖) ∧ ((𝑔‘𝑖)‘1) = (𝑦‘𝑖)))))
69	68	rspccva 3281	. . . . . . . . . . . 12 ⊢ ((∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))) ∧ 𝑖 ∈ ( I ‘𝐴)) → ((𝑔‘𝑖) ∈ (II Cn (𝐹‘𝑖)) ∧ (((𝑔‘𝑖)‘0) = (𝑥‘𝑖) ∧ ((𝑔‘𝑖)‘1) = (𝑦‘𝑖))))
70	56, 69	sylan 487	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) ∧ 𝑖 ∈ ( I ‘𝐴)) → ((𝑔‘𝑖) ∈ (II Cn (𝐹‘𝑖)) ∧ (((𝑔‘𝑖)‘0) = (𝑥‘𝑖) ∧ ((𝑔‘𝑖)‘1) = (𝑦‘𝑖))))
71	55, 70	syldan 486	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) ∧ 𝑖 ∈ 𝐴) → ((𝑔‘𝑖) ∈ (II Cn (𝐹‘𝑖)) ∧ (((𝑔‘𝑖)‘0) = (𝑥‘𝑖) ∧ ((𝑔‘𝑖)‘1) = (𝑦‘𝑖))))
72	71	simpld 474	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) ∧ 𝑖 ∈ 𝐴) → (𝑔‘𝑖) ∈ (II Cn (𝐹‘𝑖)))
73		iiuni 22492	. . . . . . . . . 10 ⊢ (0[,]1) = ∪ II
74		eqid 2610	. . . . . . . . . 10 ⊢ ∪ (𝐹‘𝑖) = ∪ (𝐹‘𝑖)
75	73, 74	cnf 20860	. . . . . . . . 9 ⊢ ((𝑔‘𝑖) ∈ (II Cn (𝐹‘𝑖)) → (𝑔‘𝑖):(0[,]1)⟶∪ (𝐹‘𝑖))
76	72, 75	syl 17	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) ∧ 𝑖 ∈ 𝐴) → (𝑔‘𝑖):(0[,]1)⟶∪ (𝐹‘𝑖))
77	76	feqmptd 6159	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) ∧ 𝑖 ∈ 𝐴) → (𝑔‘𝑖) = (𝑧 ∈ (0[,]1) ↦ ((𝑔‘𝑖)‘𝑧)))
78	77, 72	eqeltrrd 2689	. . . . . 6 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) ∧ 𝑖 ∈ 𝐴) → (𝑧 ∈ (0[,]1) ↦ ((𝑔‘𝑖)‘𝑧)) ∈ (II Cn (𝐹‘𝑖)))
79	14, 49, 50, 52, 78	ptcn 21240	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧))) ∈ (II Cn (∏t‘𝐹)))
80	71	simprd 478	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) ∧ 𝑖 ∈ 𝐴) → (((𝑔‘𝑖)‘0) = (𝑥‘𝑖) ∧ ((𝑔‘𝑖)‘1) = (𝑦‘𝑖)))
81	80	simpld 474	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) ∧ 𝑖 ∈ 𝐴) → ((𝑔‘𝑖)‘0) = (𝑥‘𝑖))
82	81	mpteq2dva 4672	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘0)) = (𝑖 ∈ 𝐴 ↦ (𝑥‘𝑖)))
83		0elunit 12161	. . . . . . 7 ⊢ 0 ∈ (0[,]1)
84		mptexg 6389	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘0)) ∈ V)
85	50, 84	syl 17	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘0)) ∈ V)
86		fveq2 6103	. . . . . . . . 9 ⊢ (𝑧 = 0 → ((𝑔‘𝑖)‘𝑧) = ((𝑔‘𝑖)‘0))
87	86	mpteq2dv 4673	. . . . . . . 8 ⊢ (𝑧 = 0 → (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)) = (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘0)))
88		eqid 2610	. . . . . . . 8 ⊢ (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧))) = (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))
89	87, 88	fvmptg 6189	. . . . . . 7 ⊢ ((0 ∈ (0[,]1) ∧ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘0)) ∈ V) → ((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘0) = (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘0)))
90	83, 85, 89	sylancr 694	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → ((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘0) = (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘0)))
91	21	simpld 474	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑥 Fn 𝐴)
92	91	adantr 480	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → 𝑥 Fn 𝐴)
93		dffn5 6151	. . . . . . 7 ⊢ (𝑥 Fn 𝐴 ↔ 𝑥 = (𝑖 ∈ 𝐴 ↦ (𝑥‘𝑖)))
94	92, 93	sylib 207	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → 𝑥 = (𝑖 ∈ 𝐴 ↦ (𝑥‘𝑖)))
95	82, 90, 94	3eqtr4d 2654	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → ((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘0) = 𝑥)
96	80	simprd 478	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) ∧ 𝑖 ∈ 𝐴) → ((𝑔‘𝑖)‘1) = (𝑦‘𝑖))
97	96	mpteq2dva 4672	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘1)) = (𝑖 ∈ 𝐴 ↦ (𝑦‘𝑖)))
98		1elunit 12162	. . . . . . 7 ⊢ 1 ∈ (0[,]1)
99		mptexg 6389	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘1)) ∈ V)
100	50, 99	syl 17	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘1)) ∈ V)
101		fveq2 6103	. . . . . . . . 9 ⊢ (𝑧 = 1 → ((𝑔‘𝑖)‘𝑧) = ((𝑔‘𝑖)‘1))
102	101	mpteq2dv 4673	. . . . . . . 8 ⊢ (𝑧 = 1 → (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)) = (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘1)))
103	102, 88	fvmptg 6189	. . . . . . 7 ⊢ ((1 ∈ (0[,]1) ∧ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘1)) ∈ V) → ((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘1) = (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘1)))
104	98, 100, 103	sylancr 694	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → ((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘1) = (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘1)))
105	28	simpld 474	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑦 Fn 𝐴)
106	105	adantr 480	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → 𝑦 Fn 𝐴)
107		dffn5 6151	. . . . . . 7 ⊢ (𝑦 Fn 𝐴 ↔ 𝑦 = (𝑖 ∈ 𝐴 ↦ (𝑦‘𝑖)))
108	106, 107	sylib 207	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → 𝑦 = (𝑖 ∈ 𝐴 ↦ (𝑦‘𝑖)))
109	97, 104, 108	3eqtr4d 2654	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → ((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘1) = 𝑦)
110		fveq1 6102	. . . . . . . 8 ⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧))) → (𝑓‘0) = ((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘0))
111	110	eqeq1d 2612	. . . . . . 7 ⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧))) → ((𝑓‘0) = 𝑥 ↔ ((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘0) = 𝑥))
112		fveq1 6102	. . . . . . . 8 ⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧))) → (𝑓‘1) = ((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘1))
113	112	eqeq1d 2612	. . . . . . 7 ⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧))) → ((𝑓‘1) = 𝑦 ↔ ((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘1) = 𝑦))
114	111, 113	anbi12d 743	. . . . . 6 ⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧))) → (((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ (((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘0) = 𝑥 ∧ ((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘1) = 𝑦)))
115	114	rspcev 3282	. . . . 5 ⊢ (((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧))) ∈ (II Cn (∏t‘𝐹)) ∧ (((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘0) = 𝑥 ∧ ((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘1) = 𝑦)) → ∃𝑓 ∈ (II Cn (∏t‘𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
116	79, 95, 109, 115	syl12anc 1316	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ∧ (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ∧ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡))))) → ∃𝑓 ∈ (II Cn (∏t‘𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
117	47, 116	exlimddv 1850	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → ∃𝑓 ∈ (II Cn (∏t‘𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
118	117	ralrimivva 2954	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) → ∀𝑥 ∈ ∪ (∏t‘𝐹)∀𝑦 ∈ ∪ (∏t‘𝐹)∃𝑓 ∈ (II Cn (∏t‘𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
119		eqid 2610	. . 3 ⊢ ∪ (∏t‘𝐹) = ∪ (∏t‘𝐹)
120	119	ispcon 30459	. 2 ⊢ ((∏t‘𝐹) ∈ PCon ↔ ((∏t‘𝐹) ∈ Top ∧ ∀𝑥 ∈ ∪ (∏t‘𝐹)∀𝑦 ∈ ∪ (∏t‘𝐹)∃𝑓 ∈ (II Cn (∏t‘𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
121	6, 118, 120	sylanbrc 695	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PCon) → (∏t‘𝐹) ∈ PCon)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∪ cuni 4372   ↦ cmpt 4643   I cid 4948   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Xcixp 7794  0cc0 9815  1c1 9816  [,]cicc 12049  ∏_tcpt 15922  Topctop 20517  TopOnctopon 20518   Cn ccn 20838  IIcii 22486  PConcpcon 30455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421  ax-ac2 9168  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-r1 8510  df-rank 8511  df-card 8648  df-ac 8822  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-icc 12053  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-topgen 15927  df-pt 15928  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cn 20841  df-cnp 20842  df-ii 22488  df-pcon 30457

This theorem is referenced by: (None)