Theorem ptpconn 35201

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 35193	. . . . 5 ⊢ (𝑥 ∈ PConn → 𝑥 ∈ Top)
2	1	ssriv 4012	. . . 4 ⊢ PConn ⊆ Top
3		fss 6763	. . . 4 ⊢ ((𝐹:𝐴⟶PConn ∧ PConn ⊆ Top) → 𝐹:𝐴⟶Top)
4	2, 3	mpan2 690	. . 3 ⊢ (𝐹:𝐴⟶PConn → 𝐹:𝐴⟶Top)
5		pttop 23611	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∏t‘𝐹) ∈ Top)
6	4, 5	sylan2 592	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) → (∏t‘𝐹) ∈ Top)
7		fvi 6998	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ‘𝐴) = 𝐴)
8	7	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → ( I ‘𝐴) = 𝐴)
9	8	eleq2d 2830	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → (𝑡 ∈ ( I ‘𝐴) ↔ 𝑡 ∈ 𝐴))
10	9	biimpa 476	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ 𝑡 ∈ ( I ‘𝐴)) → 𝑡 ∈ 𝐴)
11		simplr 768	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝐹:𝐴⟶PConn)
12	11	ffvelcdmda 7118	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ 𝑡 ∈ 𝐴) → (𝐹‘𝑡) ∈ PConn)
13		simprl 770	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑥 ∈ ∪ (∏t‘𝐹))
14		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (∏t‘𝐹) = (∏t‘𝐹)
15	14	ptuni 23623	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡) = ∪ (∏t‘𝐹))
16	4, 15	sylan2 592	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) → X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡) = ∪ (∏t‘𝐹))
17	16	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡) = ∪ (∏t‘𝐹))
18	13, 17	eleqtrrd 2847	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑥 ∈ X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡))
19		vex 3492	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
20	19	elixp 8962	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡) ↔ (𝑥 Fn 𝐴 ∧ ∀𝑡 ∈ 𝐴 (𝑥‘𝑡) ∈ ∪ (𝐹‘𝑡)))
21	18, 20	sylib 218	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → (𝑥 Fn 𝐴 ∧ ∀𝑡 ∈ 𝐴 (𝑥‘𝑡) ∈ ∪ (𝐹‘𝑡)))
22	21	simprd 495	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → ∀𝑡 ∈ 𝐴 (𝑥‘𝑡) ∈ ∪ (𝐹‘𝑡))
23	22	r19.21bi 3257	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ 𝑡 ∈ 𝐴) → (𝑥‘𝑡) ∈ ∪ (𝐹‘𝑡))
24		simprr 772	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑦 ∈ ∪ (∏t‘𝐹))
25	24, 17	eleqtrrd 2847	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑦 ∈ X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡))
26		vex 3492	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
27	26	elixp 8962	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡) ↔ (𝑦 Fn 𝐴 ∧ ∀𝑡 ∈ 𝐴 (𝑦‘𝑡) ∈ ∪ (𝐹‘𝑡)))
28	25, 27	sylib 218	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → (𝑦 Fn 𝐴 ∧ ∀𝑡 ∈ 𝐴 (𝑦‘𝑡) ∈ ∪ (𝐹‘𝑡)))
29	28	simprd 495	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → ∀𝑡 ∈ 𝐴 (𝑦‘𝑡) ∈ ∪ (𝐹‘𝑡))
30	29	r19.21bi 3257	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ 𝑡 ∈ 𝐴) → (𝑦‘𝑡) ∈ ∪ (𝐹‘𝑡))
31		eqid 2740	. . . . . . . . . 10 ⊢ ∪ (𝐹‘𝑡) = ∪ (𝐹‘𝑡)
32	31	pconncn 35192	. . . . . . . . 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 1371	. . . . . . . 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 3077	. . . . . . . 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 218	. . . . . . 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 590	. . . . . 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 3152	. . . . 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 6933	. . . . . 6 ⊢ ( I ‘𝐴) ∈ V
39		eleq1 2832	. . . . . . 7 ⊢ (𝑓 = (𝑔‘𝑡) → (𝑓 ∈ (II Cn (𝐹‘𝑡)) ↔ (𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡))))
40		fveq1 6919	. . . . . . . . 9 ⊢ (𝑓 = (𝑔‘𝑡) → (𝑓‘0) = ((𝑔‘𝑡)‘0))
41	40	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑓 = (𝑔‘𝑡) → ((𝑓‘0) = (𝑥‘𝑡) ↔ ((𝑔‘𝑡)‘0) = (𝑥‘𝑡)))
42		fveq1 6919	. . . . . . . . 9 ⊢ (𝑓 = (𝑔‘𝑡) → (𝑓‘1) = ((𝑔‘𝑡)‘1))
43	42	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑓 = (𝑔‘𝑡) → ((𝑓‘1) = (𝑦‘𝑡) ↔ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡)))
44	41, 43	anbi12d 631	. . . . . . 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 631	. . . . . 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 10555	. . . . 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⟶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 24924	. . . . . . 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 774	. . . . . 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 480	. . . . . . 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 17	. . . . . 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 480	. . . . . . . . . . . . 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 2830	. . . . . . . . . . . 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 477	. . . . . . . . . . 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 772	. . . . . . . . . . . 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 6920	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑖 → (𝑔‘𝑡) = (𝑔‘𝑖))
58		fveq2 6920	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → (𝐹‘𝑡) = (𝐹‘𝑖))
59	58	oveq2d 7464	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑖 → (II Cn (𝐹‘𝑡)) = (II Cn (𝐹‘𝑖)))
60	57, 59	eleq12d 2838	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑖 → ((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ↔ (𝑔‘𝑖) ∈ (II Cn (𝐹‘𝑖))))
61	57	fveq1d 6922	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → ((𝑔‘𝑡)‘0) = ((𝑔‘𝑖)‘0))
62		fveq2 6920	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → (𝑥‘𝑡) = (𝑥‘𝑖))
63	61, 62	eqeq12d 2756	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑖 → (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ↔ ((𝑔‘𝑖)‘0) = (𝑥‘𝑖)))
64	57	fveq1d 6922	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → ((𝑔‘𝑡)‘1) = ((𝑔‘𝑖)‘1))
65		fveq2 6920	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → (𝑦‘𝑡) = (𝑦‘𝑖))
66	64, 65	eqeq12d 2756	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑖 → (((𝑔‘𝑡)‘1) = (𝑦‘𝑡) ↔ ((𝑔‘𝑖)‘1) = (𝑦‘𝑖)))
67	63, 66	anbi12d 631	. . . . . . . . . . . . . 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 631	. . . . . . . . . . . . 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 3634	. . . . . . . . . . . 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 579	. . . . . . . . . . 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 590	. . . . . . . . . 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 494	. . . . . . . . 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 24926	. . . . . . . . . 10 ⊢ (0[,]1) = ∪ II
74		eqid 2740	. . . . . . . . . 10 ⊢ ∪ (𝐹‘𝑖) = ∪ (𝐹‘𝑖)
75	73, 74	cnf 23275	. . . . . . . . 9 ⊢ ((𝑔‘𝑖) ∈ (II Cn (𝐹‘𝑖)) → (𝑔‘𝑖):(0[,]1)⟶∪ (𝐹‘𝑖))
76	72, 75	syl 17	. . . . . . . 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 6990	. . . . . . 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 2845	. . . . . 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 23656	. . . . 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 495	. . . . . . . 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 494	. . . . . . 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 5266	. . . . . 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 13529	. . . . . . 7 ⊢ 0 ∈ (0[,]1)
84		mptexg 7258	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘0)) ∈ V)
85	50, 84	syl 17	. . . . . . 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 6920	. . . . . . . . 9 ⊢ (𝑧 = 0 → ((𝑔‘𝑖)‘𝑧) = ((𝑔‘𝑖)‘0))
87	86	mpteq2dv 5268	. . . . . . . 8 ⊢ (𝑧 = 0 → (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)) = (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘0)))
88		eqid 2740	. . . . . . . 8 ⊢ (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧))) = (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))
89	87, 88	fvmptg 7027	. . . . . . 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 586	. . . . . 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 494	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑥 Fn 𝐴)
92	91	adantr 480	. . . . . . 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 6980	. . . . . . 7 ⊢ (𝑥 Fn 𝐴 ↔ 𝑥 = (𝑖 ∈ 𝐴 ↦ (𝑥‘𝑖)))
94	92, 93	sylib 218	. . . . . 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 2790	. . . . 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 495	. . . . . . 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 5266	. . . . . 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 13530	. . . . . . 7 ⊢ 1 ∈ (0[,]1)
99		mptexg 7258	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘1)) ∈ V)
100	50, 99	syl 17	. . . . . . 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 6920	. . . . . . . . 9 ⊢ (𝑧 = 1 → ((𝑔‘𝑖)‘𝑧) = ((𝑔‘𝑖)‘1))
102	101	mpteq2dv 5268	. . . . . . . 8 ⊢ (𝑧 = 1 → (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)) = (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘1)))
103	102, 88	fvmptg 7027	. . . . . . 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 586	. . . . . 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 494	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑦 Fn 𝐴)
106	105	adantr 480	. . . . . . 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 6980	. . . . . . 7 ⊢ (𝑦 Fn 𝐴 ↔ 𝑦 = (𝑖 ∈ 𝐴 ↦ (𝑦‘𝑖)))
108	106, 107	sylib 218	. . . . . 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 2790	. . . . 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 6919	. . . . . . . 8 ⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧))) → (𝑓‘0) = ((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘0))
111	110	eqeq1d 2742	. . . . . . 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 6919	. . . . . . . 8 ⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧))) → (𝑓‘1) = ((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘1))
113	112	eqeq1d 2742	. . . . . . 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 631	. . . . . 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 3635	. . . . 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 836	. . . 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 1934	. . 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 3208	. 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 2740	. . 3 ⊢ ∪ (∏t‘𝐹) = ∪ (∏t‘𝐹)
120	119	ispconn 35191	. 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 582	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) → (∏t‘𝐹) ∈ PConn)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ⊆ wss 3976  ∪ cuni 4931   ↦ cmpt 5249   I cid 5592   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  Xcixp 8955  0cc0 11184  1c1 11185  [,]cicc 13410  ∏_tcpt 17498  Topctop 22920  TopOnctopon 22937   Cn ccn 23253  IIcii 24920  PConncpconn 35187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-reg 9661  ax-inf2 9710  ax-ac2 10532  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fi 9480  df-sup 9511  df-inf 9512  df-r1 9833  df-rank 9834  df-card 10008  df-ac 10185  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-q 13014  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-icc 13414  df-seq 14053  df-exp 14113  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-topgen 17503  df-pt 17504  df-psmet 21379  df-xmet 21380  df-met 21381  df-bl 21382  df-mopn 21383  df-top 22921  df-topon 22938  df-bases 22974  df-cn 23256  df-cnp 23257  df-ii 24922  df-pconn 35189

This theorem is referenced by: (None)