Theorem ptpconn 35998

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
ptpconn	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) → (∏t‘𝐹) ∈ PConn)

Proof of Theorem ptpconn

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

Step	Hyp	Ref	Expression
1		pconntop 35990	. . . . 5 ⊢ (𝑥 ∈ PConn → 𝑥 ∈ Top)
2	1	ssriv 3912	. . . 4 ⊢ PConn ⊆ Top
3		fss 6726	. . . 4 ⊢ ((𝐹:𝐴⟶PConn ∧ PConn ⊆ Top) → 𝐹:𝐴⟶Top)
4	2, 3	mpan2 704	. . 3 ⊢ (𝐹:𝐴⟶PConn → 𝐹:𝐴⟶Top)
5		pttop 23901	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∏t‘𝐹) ∈ Top)
6	4, 5	sylan2 605	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) → (∏t‘𝐹) ∈ Top)
7		fvi 6961	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ‘𝐴) = 𝐴)
8	7	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → ( I ‘𝐴) = 𝐴)
9	8	eleq2d 2846	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → (𝑡 ∈ ( I ‘𝐴) ↔ 𝑡 ∈ 𝐴))
10	9	biimpa 482	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ 𝑡 ∈ ( I ‘𝐴)) → 𝑡 ∈ 𝐴)
11		simplr 781	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝐹:𝐴⟶PConn)
12	11	ffvelcdmda 7084	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ 𝑡 ∈ 𝐴) → (𝐹‘𝑡) ∈ PConn)
13		simprl 783	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑥 ∈ ∪ (∏t‘𝐹))
14		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (∏t‘𝐹) = (∏t‘𝐹)
15	14	ptuni 23913	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡) = ∪ (∏t‘𝐹))
16	4, 15	sylan2 605	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) → X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡) = ∪ (∏t‘𝐹))
17	16	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡) = ∪ (∏t‘𝐹))
18	13, 17	eleqtrrd 2863	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑥 ∈ X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡))
19		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
20	19	elixp 8932	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡) ↔ (𝑥 Fn 𝐴 ∧ ∀𝑡 ∈ 𝐴 (𝑥‘𝑡) ∈ ∪ (𝐹‘𝑡)))
21	18, 20	sylib 221	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → (𝑥 Fn 𝐴 ∧ ∀𝑡 ∈ 𝐴 (𝑥‘𝑡) ∈ ∪ (𝐹‘𝑡)))
22	21	simprd 501	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → ∀𝑡 ∈ 𝐴 (𝑥‘𝑡) ∈ ∪ (𝐹‘𝑡))
23	22	r19.21bi 3254	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ 𝑡 ∈ 𝐴) → (𝑥‘𝑡) ∈ ∪ (𝐹‘𝑡))
24		simprr 785	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑦 ∈ ∪ (∏t‘𝐹))
25	24, 17	eleqtrrd 2863	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑦 ∈ X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡))
26		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
27	26	elixp 8932	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡) ↔ (𝑦 Fn 𝐴 ∧ ∀𝑡 ∈ 𝐴 (𝑦‘𝑡) ∈ ∪ (𝐹‘𝑡)))
28	25, 27	sylib 221	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → (𝑦 Fn 𝐴 ∧ ∀𝑡 ∈ 𝐴 (𝑦‘𝑡) ∈ ∪ (𝐹‘𝑡)))
29	28	simprd 501	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → ∀𝑡 ∈ 𝐴 (𝑦‘𝑡) ∈ ∪ (𝐹‘𝑡))
30	29	r19.21bi 3254	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ 𝑡 ∈ 𝐴) → (𝑦‘𝑡) ∈ ∪ (𝐹‘𝑡))
31		eqid 2760	. . . . . . . . . 10 ⊢ ∪ (𝐹‘𝑡) = ∪ (𝐹‘𝑡)
32	31	pconncn 35989	. . . . . . . . 9 ⊢ (((𝐹‘𝑡) ∈ PConn ∧ (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 1398	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ 𝑡 ∈ 𝐴) → ∃𝑓 ∈ (II Cn (𝐹‘𝑡))((𝑓‘0) = (𝑥‘𝑡) ∧ (𝑓‘1) = (𝑦‘𝑡)))
34		df-rex 3087	. . . . . . . 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 221	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 603	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 3154	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 6898	. . . . . 6 ⊢ ( I ‘𝐴) ∈ V
39		eleq1 2848	. . . . . . 7 ⊢ (𝑓 = (𝑔‘𝑡) → (𝑓 ∈ (II Cn (𝐹‘𝑡)) ↔ (𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡))))
40		fveq1 6884	. . . . . . . . 9 ⊢ (𝑓 = (𝑔‘𝑡) → (𝑓‘0) = ((𝑔‘𝑡)‘0))
41	40	eqeq1d 2762	. . . . . . . 8 ⊢ (𝑓 = (𝑔‘𝑡) → ((𝑓‘0) = (𝑥‘𝑡) ↔ ((𝑔‘𝑡)‘0) = (𝑥‘𝑡)))
42		fveq1 6884	. . . . . . . . 9 ⊢ (𝑓 = (𝑔‘𝑡) → (𝑓‘1) = ((𝑔‘𝑡)‘1))
43	42	eqeq1d 2762	. . . . . . . 8 ⊢ (𝑓 = (𝑔‘𝑡) → ((𝑓‘1) = (𝑦‘𝑡) ↔ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡)))
44	41, 43	anbi12d 644	. . . . . . 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 644	. . . . . 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 10564	. . . . 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 18	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 25200	. . . . . . 7 ⊢ II ∈ (TopOn‘(0[,]1))
49	48	a1i 11	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 787	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 486	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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⟶PConn)
52	51, 4	syl 18	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 486	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 2846	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 483	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 785	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑖 → (𝑔‘𝑡) = (𝑔‘𝑖))
58		fveq2 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → (𝐹‘𝑡) = (𝐹‘𝑖))
59	58	oveq2d 7436	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑖 → (II Cn (𝐹‘𝑡)) = (II Cn (𝐹‘𝑖)))
60	57, 59	eleq12d 2854	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑖 → ((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ↔ (𝑔‘𝑖) ∈ (II Cn (𝐹‘𝑖))))
61	57	fveq1d 6887	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → ((𝑔‘𝑡)‘0) = ((𝑔‘𝑖)‘0))
62		fveq2 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → (𝑥‘𝑡) = (𝑥‘𝑖))
63	61, 62	eqeq12d 2776	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑖 → (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ↔ ((𝑔‘𝑖)‘0) = (𝑥‘𝑖)))
64	57	fveq1d 6887	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → ((𝑔‘𝑡)‘1) = ((𝑔‘𝑖)‘1))
65		fveq2 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → (𝑦‘𝑡) = (𝑦‘𝑖))
66	64, 65	eqeq12d 2776	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑖 → (((𝑔‘𝑡)‘1) = (𝑦‘𝑡) ↔ ((𝑔‘𝑖)‘1) = (𝑦‘𝑖)))
67	63, 66	anbi12d 644	. . . . . . . . . . . . . 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 644	. . . . . . . . . . . . 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 3575	. . . . . . . . . . . 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 592	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 603	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 500	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 25202	. . . . . . . . . 10 ⊢ (0[,]1) = ∪ II
74		eqid 2760	. . . . . . . . . 10 ⊢ ∪ (𝐹‘𝑖) = ∪ (𝐹‘𝑖)
75	73, 74	cnf 23564	. . . . . . . . 9 ⊢ ((𝑔‘𝑖) ∈ (II Cn (𝐹‘𝑖)) → (𝑔‘𝑖):(0[,]1)⟶∪ (𝐹‘𝑖))
76	72, 75	syl 18	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 6953	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 2861	. . . . . 6 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 23946	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 501	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 500	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 5197	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 13600	. . . . . . 7 ⊢ 0 ∈ (0[,]1)
84		mptexg 7227	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘0)) ∈ V)
85	50, 84	syl 18	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 6885	. . . . . . . . 9 ⊢ (𝑧 = 0 → ((𝑔‘𝑖)‘𝑧) = ((𝑔‘𝑖)‘0))
87	86	mpteq2dv 5198	. . . . . . . 8 ⊢ (𝑧 = 0 → (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)) = (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘0)))
88		eqid 2760	. . . . . . . 8 ⊢ (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧))) = (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))
89	87, 88	fvmptg 6991	. . . . . . 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 599	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 500	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑥 Fn 𝐴)
92	91	adantr 486	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 6943	. . . . . . 7 ⊢ (𝑥 Fn 𝐴 ↔ 𝑥 = (𝑖 ∈ 𝐴 ↦ (𝑥‘𝑖)))
94	92, 93	sylib 221	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 2805	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 501	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 5197	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 13601	. . . . . . 7 ⊢ 1 ∈ (0[,]1)
99		mptexg 7227	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘1)) ∈ V)
100	50, 99	syl 18	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 6885	. . . . . . . . 9 ⊢ (𝑧 = 1 → ((𝑔‘𝑖)‘𝑧) = ((𝑔‘𝑖)‘1))
102	101	mpteq2dv 5198	. . . . . . . 8 ⊢ (𝑧 = 1 → (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)) = (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘1)))
103	102, 88	fvmptg 6991	. . . . . . 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 599	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 500	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑦 Fn 𝐴)
106	105	adantr 486	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 6943	. . . . . . 7 ⊢ (𝑦 Fn 𝐴 ↔ 𝑦 = (𝑖 ∈ 𝐴 ↦ (𝑦‘𝑖)))
108	106, 107	sylib 221	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 2805	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 6884	. . . . . . . 8 ⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧))) → (𝑓‘0) = ((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘0))
111	110	eqeq1d 2762	. . . . . . 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 6884	. . . . . . . 8 ⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧))) → (𝑓‘1) = ((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘1))
113	112	eqeq1d 2762	. . . . . . 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 644	. . . . . 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 3576	. . . . 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 850	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (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 1967	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → ∃𝑓 ∈ (II Cn (∏t‘𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
118	117	ralrimivva 3205	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) → ∀𝑥 ∈ ∪ (∏t‘𝐹)∀𝑦 ∈ ∪ (∏t‘𝐹)∃𝑓 ∈ (II Cn (∏t‘𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
119		eqid 2760	. . 3 ⊢ ∪ (∏t‘𝐹) = ∪ (∏t‘𝐹)
120	119	ispconn 35988	. 2 ⊢ ((∏t‘𝐹) ∈ PConn ↔ ((∏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 595	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) → (∏t‘𝐹) ∈ PConn)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895  ∪ cuni 4866   ↦ cmpt 5185   I cid 5544   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  Xcixp 8925  0cc0 11200  1c1 11201  [,]cicc 13479  ∏_tcpt 17609  Topctop 23211  TopOnctopon 23228   Cn ccn 23542  IIcii 25196  PConncpconn 35984

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-reg 9586  ax-inf2 9642  ax-ac2 10541  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-map 8849  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-fi 9403  df-sup 9434  df-inf 9435  df-r1 9768  df-rank 9769  df-scott 9929  df-card 10020  df-ac 10195  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-n0 12607  df-z 12694  df-uz 12966  df-q 13076  df-rp 13121  df-xneg 13241  df-xadd 13242  df-xmul 13243  df-icc 13483  df-seq 14145  df-exp 14205  df-cj 15266  df-re 15267  df-im 15268  df-sqrt 15402  df-abs 15403  df-topgen 17614  df-pt 17615  df-psmet 21670  df-xmet 21671  df-met 21672  df-bl 21673  df-mopn 21674  df-top 23212  df-topon 23229  df-bases 23264  df-cn 23545  df-cnp 23546  df-ii 25198  df-pconn 35986

This theorem is used by: (None)