Theorem ptpconn 32593

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 32585	. . . . 5 ⊢ (𝑥 ∈ PConn → 𝑥 ∈ Top)
2	1	ssriv 3919	. . . 4 ⊢ PConn ⊆ Top
3		fss 6501	. . . 4 ⊢ ((𝐹:𝐴⟶PConn ∧ PConn ⊆ Top) → 𝐹:𝐴⟶Top)
4	2, 3	mpan2 690	. . 3 ⊢ (𝐹:𝐴⟶PConn → 𝐹:𝐴⟶Top)
5		pttop 22187	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∏t‘𝐹) ∈ Top)
6	4, 5	sylan2 595	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) → (∏t‘𝐹) ∈ Top)
7		fvi 6715	. . . . . . . . . 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 2875	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → (𝑡 ∈ ( I ‘𝐴) ↔ 𝑡 ∈ 𝐴))
10	9	biimpa 480	. . . . . . 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	ffvelrnda 6828	. . . . . . . . 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 2798	. . . . . . . . . . . . . . . 16 ⊢ (∏t‘𝐹) = (∏t‘𝐹)
15	14	ptuni 22199	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡) = ∪ (∏t‘𝐹))
16	4, 15	sylan2 595	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) → X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡) = ∪ (∏t‘𝐹))
17	16	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡) = ∪ (∏t‘𝐹))
18	13, 17	eleqtrrd 2893	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑥 ∈ X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡))
19		vex 3444	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
20	19	elixp 8451	. . . . . . . . . . . 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 499	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → ∀𝑡 ∈ 𝐴 (𝑥‘𝑡) ∈ ∪ (𝐹‘𝑡))
23	22	r19.21bi 3173	. . . . . . . . 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 2893	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑦 ∈ X𝑡 ∈ 𝐴 ∪ (𝐹‘𝑡))
26		vex 3444	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
27	26	elixp 8451	. . . . . . . . . . . 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 499	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → ∀𝑡 ∈ 𝐴 (𝑦‘𝑡) ∈ ∪ (𝐹‘𝑡))
30	29	r19.21bi 3173	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) ∧ 𝑡 ∈ 𝐴) → (𝑦‘𝑡) ∈ ∪ (𝐹‘𝑡))
31		eqid 2798	. . . . . . . . . 10 ⊢ ∪ (𝐹‘𝑡) = ∪ (𝐹‘𝑡)
32	31	pconncn 32584	. . . . . . . . 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 1368	. . . . . . . 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 3112	. . . . . . . 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 594	. . . . . 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 3149	. . . . 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 6658	. . . . . 6 ⊢ ( I ‘𝐴) ∈ V
39		eleq1 2877	. . . . . . 7 ⊢ (𝑓 = (𝑔‘𝑡) → (𝑓 ∈ (II Cn (𝐹‘𝑡)) ↔ (𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡))))
40		fveq1 6644	. . . . . . . . 9 ⊢ (𝑓 = (𝑔‘𝑡) → (𝑓‘0) = ((𝑔‘𝑡)‘0))
41	40	eqeq1d 2800	. . . . . . . 8 ⊢ (𝑓 = (𝑔‘𝑡) → ((𝑓‘0) = (𝑥‘𝑡) ↔ ((𝑔‘𝑡)‘0) = (𝑥‘𝑡)))
42		fveq1 6644	. . . . . . . . 9 ⊢ (𝑓 = (𝑔‘𝑡) → (𝑓‘1) = ((𝑔‘𝑡)‘1))
43	42	eqeq1d 2800	. . . . . . . 8 ⊢ (𝑓 = (𝑔‘𝑡) → ((𝑓‘1) = (𝑦‘𝑡) ↔ ((𝑔‘𝑡)‘1) = (𝑦‘𝑡)))
44	41, 43	anbi12d 633	. . . . . . 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 633	. . . . . 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 9897	. . . . 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 23484	. . . . . . 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 484	. . . . . . 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 484	. . . . . . . . . . . . 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 2875	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 6645	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑖 → (𝑔‘𝑡) = (𝑔‘𝑖))
58		fveq2 6645	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → (𝐹‘𝑡) = (𝐹‘𝑖))
59	58	oveq2d 7151	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑖 → (II Cn (𝐹‘𝑡)) = (II Cn (𝐹‘𝑖)))
60	57, 59	eleq12d 2884	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑖 → ((𝑔‘𝑡) ∈ (II Cn (𝐹‘𝑡)) ↔ (𝑔‘𝑖) ∈ (II Cn (𝐹‘𝑖))))
61	57	fveq1d 6647	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → ((𝑔‘𝑡)‘0) = ((𝑔‘𝑖)‘0))
62		fveq2 6645	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → (𝑥‘𝑡) = (𝑥‘𝑖))
63	61, 62	eqeq12d 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑖 → (((𝑔‘𝑡)‘0) = (𝑥‘𝑡) ↔ ((𝑔‘𝑖)‘0) = (𝑥‘𝑖)))
64	57	fveq1d 6647	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → ((𝑔‘𝑡)‘1) = ((𝑔‘𝑖)‘1))
65		fveq2 6645	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑖 → (𝑦‘𝑡) = (𝑦‘𝑖))
66	64, 65	eqeq12d 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑖 → (((𝑔‘𝑡)‘1) = (𝑦‘𝑡) ↔ ((𝑔‘𝑖)‘1) = (𝑦‘𝑖)))
67	63, 66	anbi12d 633	. . . . . . . . . . . . . 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 633	. . . . . . . . . . . . 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 3570	. . . . . . . . . . . 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 583	. . . . . . . . . . 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 594	. . . . . . . . . 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 498	. . . . . . . . 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 23486	. . . . . . . . . 10 ⊢ (0[,]1) = ∪ II
74		eqid 2798	. . . . . . . . . 10 ⊢ ∪ (𝐹‘𝑖) = ∪ (𝐹‘𝑖)
75	73, 74	cnf 21851	. . . . . . . . 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 6708	. . . . . . 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 2891	. . . . . 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 22232	. . . . 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 499	. . . . . . . 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 498	. . . . . . 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 5125	. . . . . 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 12847	. . . . . . 7 ⊢ 0 ∈ (0[,]1)
84		mptexg 6961	. . . . . . . 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 6645	. . . . . . . . 9 ⊢ (𝑧 = 0 → ((𝑔‘𝑖)‘𝑧) = ((𝑔‘𝑖)‘0))
87	86	mpteq2dv 5126	. . . . . . . 8 ⊢ (𝑧 = 0 → (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)) = (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘0)))
88		eqid 2798	. . . . . . . 8 ⊢ (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧))) = (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))
89	87, 88	fvmptg 6743	. . . . . . 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 590	. . . . . 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 498	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑥 Fn 𝐴)
92	91	adantr 484	. . . . . . 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 6699	. . . . . . 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 2843	. . . . 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 499	. . . . . . 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 5125	. . . . . 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 12848	. . . . . . 7 ⊢ 1 ∈ (0[,]1)
99		mptexg 6961	. . . . . . . 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 6645	. . . . . . . . 9 ⊢ (𝑧 = 1 → ((𝑔‘𝑖)‘𝑧) = ((𝑔‘𝑖)‘1))
102	101	mpteq2dv 5126	. . . . . . . 8 ⊢ (𝑧 = 1 → (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)) = (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘1)))
103	102, 88	fvmptg 6743	. . . . . . 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 590	. . . . . 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 498	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) ∧ (𝑥 ∈ ∪ (∏t‘𝐹) ∧ 𝑦 ∈ ∪ (∏t‘𝐹))) → 𝑦 Fn 𝐴)
106	105	adantr 484	. . . . . . 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 6699	. . . . . . 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 2843	. . . . 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 6644	. . . . . . . 8 ⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧))) → (𝑓‘0) = ((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘0))
111	110	eqeq1d 2800	. . . . . . 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 6644	. . . . . . . 8 ⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧))) → (𝑓‘1) = ((𝑧 ∈ (0[,]1) ↦ (𝑖 ∈ 𝐴 ↦ ((𝑔‘𝑖)‘𝑧)))‘1))
113	112	eqeq1d 2800	. . . . . . 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 633	. . . . . 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 3571	. . . . 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 835	. . . 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 1936	. . 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 3156	. 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 2798	. . 3 ⊢ ∪ (∏t‘𝐹) = ∪ (∏t‘𝐹)
120	119	ispconn 32583	. 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 586	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) → (∏t‘𝐹) ∈ PConn)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881  ∪ cuni 4800   ↦ cmpt 5110   I cid 5424   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  Xcixp 8444  0cc0 10526  1c1 10527  [,]cicc 12729  ∏_tcpt 16704  Topctop 21498  TopOnctopon 21515   Cn ccn 21829  IIcii 23480  PConncpconn 32579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-reg 9040  ax-inf2 9088  ax-ac2 9874  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fi 8859  df-sup 8890  df-inf 8891  df-r1 9177  df-rank 9178  df-card 9352  df-ac 9527  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-icc 12733  df-seq 13365  df-exp 13426  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-topgen 16709  df-pt 16710  df-psmet 20083  df-xmet 20084  df-met 20085  df-bl 20086  df-mopn 20087  df-top 21499  df-topon 21516  df-bases 21551  df-cn 21832  df-cnp 21833  df-ii 23482  df-pconn 32581

This theorem is referenced by: (None)