Theorem txsconnlem 31222

Description: Lemma for txsconn 31223. (Contributed by Mario Carneiro, 9-Mar-2015.)

Hypotheses

Ref	Expression
txsconn.1	⊢ (𝜑 → 𝑅 ∈ Top)
txsconn.2	⊢ (𝜑 → 𝑆 ∈ Top)
txsconn.3	⊢ (𝜑 → 𝐹 ∈ (II Cn (𝑅 ×t 𝑆)))
txsconn.5	⊢ 𝐴 = ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)
txsconn.6	⊢ 𝐵 = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)
txsconn.7	⊢ (𝜑 → 𝐺 ∈ (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)})))
txsconn.8	⊢ (𝜑 → 𝐻 ∈ (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)})))

Assertion

Ref	Expression
txsconnlem	⊢ (𝜑 → 𝐹( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)}))

Proof of Theorem txsconnlem

Dummy variables 𝑥 𝑠 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		txsconn.3	. 2 ⊢ (𝜑 → 𝐹 ∈ (II Cn (𝑅 ×t 𝑆)))
2		fconstmpt 5163	. . 3 ⊢ ((0[,]1) × {(𝐹‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐹‘0))
3		iitopon 22682	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
4	3	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
5		txsconn.1	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Top)
6		eqid 2622	. . . . . . 7 ⊢ ∪ 𝑅 = ∪ 𝑅
7	6	toptopon 20722	. . . . . 6 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅))
8	5, 7	sylib 208	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (TopOn‘∪ 𝑅))
9		txsconn.2	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Top)
10		eqid 2622	. . . . . . 7 ⊢ ∪ 𝑆 = ∪ 𝑆
11	10	toptopon 20722	. . . . . 6 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘∪ 𝑆))
12	9, 11	sylib 208	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (TopOn‘∪ 𝑆))
13		txtopon 21394	. . . . 5 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
14	8, 12, 13	syl2anc 693	. . . 4 ⊢ (𝜑 → (𝑅 ×t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
15		cnf2 21053	. . . . . 6 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ (𝑅 ×t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)) ∧ 𝐹 ∈ (II Cn (𝑅 ×t 𝑆))) → 𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆))
16	4, 14, 1, 15	syl3anc 1326	. . . . 5 ⊢ (𝜑 → 𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆))
17		0elunit 12290	. . . . 5 ⊢ 0 ∈ (0[,]1)
18		ffvelrn 6357	. . . . 5 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ (∪ 𝑅 × ∪ 𝑆))
19	16, 17, 18	sylancl 694	. . . 4 ⊢ (𝜑 → (𝐹‘0) ∈ (∪ 𝑅 × ∪ 𝑆))
20	4, 14, 19	cnmptc 21465	. . 3 ⊢ (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐹‘0)) ∈ (II Cn (𝑅 ×t 𝑆)))
21	2, 20	syl5eqel 2705	. 2 ⊢ (𝜑 → ((0[,]1) × {(𝐹‘0)}) ∈ (II Cn (𝑅 ×t 𝑆)))
22		txsconn.5	. . . . . . . . . . 11 ⊢ 𝐴 = ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)
23		tx1cn 21412	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
24	8, 12, 23	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
25		cnco 21070	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅)) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑅))
26	1, 24, 25	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑅))
27	22, 26	syl5eqel 2705	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (II Cn 𝑅))
28		fconstmpt 5163	. . . . . . . . . . 11 ⊢ ((0[,]1) × {(𝐴‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐴‘0))
29		iiuni 22684	. . . . . . . . . . . . . . 15 ⊢ (0[,]1) = ∪ II
30	29, 6	cnf 21050	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (II Cn 𝑅) → 𝐴:(0[,]1)⟶∪ 𝑅)
31	27, 30	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴:(0[,]1)⟶∪ 𝑅)
32		ffvelrn 6357	. . . . . . . . . . . . 13 ⊢ ((𝐴:(0[,]1)⟶∪ 𝑅 ∧ 0 ∈ (0[,]1)) → (𝐴‘0) ∈ ∪ 𝑅)
33	31, 17, 32	sylancl 694	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴‘0) ∈ ∪ 𝑅)
34	4, 8, 33	cnmptc 21465	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐴‘0)) ∈ (II Cn 𝑅))
35	28, 34	syl5eqel 2705	. . . . . . . . . 10 ⊢ (𝜑 → ((0[,]1) × {(𝐴‘0)}) ∈ (II Cn 𝑅))
36	27, 35	phtpycn 22782	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)})) ⊆ ((II ×t II) Cn 𝑅))
37		txsconn.7	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)})))
38	36, 37	sseldd 3604	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝑅))
39		iitop 22683	. . . . . . . . . 10 ⊢ II ∈ Top
40	39, 39, 29, 29	txunii 21396	. . . . . . . . 9 ⊢ ((0[,]1) × (0[,]1)) = ∪ (II ×t II)
41	40, 6	cnf 21050	. . . . . . . 8 ⊢ (𝐺 ∈ ((II ×t II) Cn 𝑅) → 𝐺:((0[,]1) × (0[,]1))⟶∪ 𝑅)
42		ffn 6045	. . . . . . . 8 ⊢ (𝐺:((0[,]1) × (0[,]1))⟶∪ 𝑅 → 𝐺 Fn ((0[,]1) × (0[,]1)))
43	38, 41, 42	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ((0[,]1) × (0[,]1)))
44		fnov 6768	. . . . . . 7 ⊢ (𝐺 Fn ((0[,]1) × (0[,]1)) ↔ 𝐺 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐺𝑦)))
45	43, 44	sylib 208	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐺𝑦)))
46	45, 38	eqeltrrd 2702	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐺𝑦)) ∈ ((II ×t II) Cn 𝑅))
47		txsconn.6	. . . . . . . . . . 11 ⊢ 𝐵 = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)
48		tx2cn 21413	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
49	8, 12, 48	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → (2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
50		cnco 21070	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆)) → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑆))
51	1, 49, 50	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑆))
52	47, 51	syl5eqel 2705	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ (II Cn 𝑆))
53		fconstmpt 5163	. . . . . . . . . . 11 ⊢ ((0[,]1) × {(𝐵‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐵‘0))
54	29, 10	cnf 21050	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (II Cn 𝑆) → 𝐵:(0[,]1)⟶∪ 𝑆)
55	52, 54	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵:(0[,]1)⟶∪ 𝑆)
56		ffvelrn 6357	. . . . . . . . . . . . 13 ⊢ ((𝐵:(0[,]1)⟶∪ 𝑆 ∧ 0 ∈ (0[,]1)) → (𝐵‘0) ∈ ∪ 𝑆)
57	55, 17, 56	sylancl 694	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵‘0) ∈ ∪ 𝑆)
58	4, 12, 57	cnmptc 21465	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐵‘0)) ∈ (II Cn 𝑆))
59	53, 58	syl5eqel 2705	. . . . . . . . . 10 ⊢ (𝜑 → ((0[,]1) × {(𝐵‘0)}) ∈ (II Cn 𝑆))
60	52, 59	phtpycn 22782	. . . . . . . . 9 ⊢ (𝜑 → (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)})) ⊆ ((II ×t II) Cn 𝑆))
61		txsconn.8	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)})))
62	60, 61	sseldd 3604	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ ((II ×t II) Cn 𝑆))
63	40, 10	cnf 21050	. . . . . . . 8 ⊢ (𝐻 ∈ ((II ×t II) Cn 𝑆) → 𝐻:((0[,]1) × (0[,]1))⟶∪ 𝑆)
64		ffn 6045	. . . . . . . 8 ⊢ (𝐻:((0[,]1) × (0[,]1))⟶∪ 𝑆 → 𝐻 Fn ((0[,]1) × (0[,]1)))
65	62, 63, 64	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐻 Fn ((0[,]1) × (0[,]1)))
66		fnov 6768	. . . . . . 7 ⊢ (𝐻 Fn ((0[,]1) × (0[,]1)) ↔ 𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)))
67	65, 66	sylib 208	. . . . . 6 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)))
68	67, 62	eqeltrrd 2702	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)) ∈ ((II ×t II) Cn 𝑆))
69	4, 4, 46, 68	cnmpt2t 21476	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) ∈ ((II ×t II) Cn (𝑅 ×t 𝑆)))
70	27, 35	phtpyhtpy 22781	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)})) ⊆ (𝐴(II Htpy 𝑅)((0[,]1) × {(𝐴‘0)})))
71	70, 37	sseldd 3604	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (𝐴(II Htpy 𝑅)((0[,]1) × {(𝐴‘0)})))
72	4, 27, 35, 71	htpyi 22773	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝑠𝐺0) = (𝐴‘𝑠) ∧ (𝑠𝐺1) = (((0[,]1) × {(𝐴‘0)})‘𝑠)))
73	72	simpld 475	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐺0) = (𝐴‘𝑠))
74	22	fveq1i 6192	. . . . . . . 8 ⊢ (𝐴‘𝑠) = (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘𝑠)
75		fvco3 6275	. . . . . . . . 9 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘𝑠) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)))
76	16, 75	sylan 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘𝑠) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)))
77	74, 76	syl5eq 2668	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐴‘𝑠) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)))
78		ffvelrn 6357	. . . . . . . . 9 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘𝑠) ∈ (∪ 𝑅 × ∪ 𝑆))
79	16, 78	sylan 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘𝑠) ∈ (∪ 𝑅 × ∪ 𝑆))
80		fvres 6207	. . . . . . . 8 ⊢ ((𝐹‘𝑠) ∈ (∪ 𝑅 × ∪ 𝑆) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)) = (1st ‘(𝐹‘𝑠)))
81	79, 80	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)) = (1st ‘(𝐹‘𝑠)))
82	73, 77, 81	3eqtrd 2660	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐺0) = (1st ‘(𝐹‘𝑠)))
83	52, 59	phtpyhtpy 22781	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)})) ⊆ (𝐵(II Htpy 𝑆)((0[,]1) × {(𝐵‘0)})))
84	83, 61	sseldd 3604	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ (𝐵(II Htpy 𝑆)((0[,]1) × {(𝐵‘0)})))
85	4, 52, 59, 84	htpyi 22773	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝑠𝐻0) = (𝐵‘𝑠) ∧ (𝑠𝐻1) = (((0[,]1) × {(𝐵‘0)})‘𝑠)))
86	85	simpld 475	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐵‘𝑠))
87	47	fveq1i 6192	. . . . . . . 8 ⊢ (𝐵‘𝑠) = (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘𝑠)
88		fvco3 6275	. . . . . . . . 9 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘𝑠) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)))
89	16, 88	sylan 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘𝑠) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)))
90	87, 89	syl5eq 2668	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐵‘𝑠) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)))
91		fvres 6207	. . . . . . . 8 ⊢ ((𝐹‘𝑠) ∈ (∪ 𝑅 × ∪ 𝑆) → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)) = (2nd ‘(𝐹‘𝑠)))
92	79, 91	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)) = (2nd ‘(𝐹‘𝑠)))
93	86, 90, 92	3eqtrd 2660	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (2nd ‘(𝐹‘𝑠)))
94	82, 93	opeq12d 4410	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ⟨(𝑠𝐺0), (𝑠𝐻0)⟩ = ⟨(1st ‘(𝐹‘𝑠)), (2nd ‘(𝐹‘𝑠))⟩)
95		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ (0[,]1))
96		oveq12 6659	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑥𝐺𝑦) = (𝑠𝐺0))
97		oveq12 6659	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑥𝐻𝑦) = (𝑠𝐻0))
98	96, 97	opeq12d 4410	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
99		eqid 2622	. . . . . . 7 ⊢ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)
100		opex 4932	. . . . . . 7 ⊢ ⟨(𝑠𝐺0), (𝑠𝐻0)⟩ ∈ V
101	98, 99, 100	ovmpt2a 6791	. . . . . 6 ⊢ ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
102	95, 17, 101	sylancl 694	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
103		1st2nd2 7205	. . . . . 6 ⊢ ((𝐹‘𝑠) ∈ (∪ 𝑅 × ∪ 𝑆) → (𝐹‘𝑠) = ⟨(1st ‘(𝐹‘𝑠)), (2nd ‘(𝐹‘𝑠))⟩)
104	79, 103	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘𝑠) = ⟨(1st ‘(𝐹‘𝑠)), (2nd ‘(𝐹‘𝑠))⟩)
105	94, 102, 104	3eqtr4d 2666	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = (𝐹‘𝑠))
106	72	simprd 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐺1) = (((0[,]1) × {(𝐴‘0)})‘𝑠))
107		fvex 6201	. . . . . . . . 9 ⊢ (𝐴‘0) ∈ V
108	107	fvconst2 6469	. . . . . . . 8 ⊢ (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝐴‘0)})‘𝑠) = (𝐴‘0))
109	108	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐴‘0)})‘𝑠) = (𝐴‘0))
110	22	fveq1i 6192	. . . . . . . . 9 ⊢ (𝐴‘0) = (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0)
111		fvco3 6275	. . . . . . . . . . 11 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 0 ∈ (0[,]1)) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)))
112	16, 17, 111	sylancl 694	. . . . . . . . . 10 ⊢ (𝜑 → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)))
113		fvres 6207	. . . . . . . . . . 11 ⊢ ((𝐹‘0) ∈ (∪ 𝑅 × ∪ 𝑆) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)) = (1st ‘(𝐹‘0)))
114	19, 113	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)) = (1st ‘(𝐹‘0)))
115	112, 114	eqtrd 2656	. . . . . . . . 9 ⊢ (𝜑 → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0) = (1st ‘(𝐹‘0)))
116	110, 115	syl5eq 2668	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘0) = (1st ‘(𝐹‘0)))
117	116	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐴‘0) = (1st ‘(𝐹‘0)))
118	106, 109, 117	3eqtrd 2660	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐺1) = (1st ‘(𝐹‘0)))
119	85	simprd 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (((0[,]1) × {(𝐵‘0)})‘𝑠))
120		fvex 6201	. . . . . . . . 9 ⊢ (𝐵‘0) ∈ V
121	120	fvconst2 6469	. . . . . . . 8 ⊢ (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝐵‘0)})‘𝑠) = (𝐵‘0))
122	121	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐵‘0)})‘𝑠) = (𝐵‘0))
123	47	fveq1i 6192	. . . . . . . . 9 ⊢ (𝐵‘0) = (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0)
124		fvco3 6275	. . . . . . . . . . 11 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 0 ∈ (0[,]1)) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)))
125	16, 17, 124	sylancl 694	. . . . . . . . . 10 ⊢ (𝜑 → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)))
126		fvres 6207	. . . . . . . . . . 11 ⊢ ((𝐹‘0) ∈ (∪ 𝑅 × ∪ 𝑆) → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)) = (2nd ‘(𝐹‘0)))
127	19, 126	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)) = (2nd ‘(𝐹‘0)))
128	125, 127	eqtrd 2656	. . . . . . . . 9 ⊢ (𝜑 → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0) = (2nd ‘(𝐹‘0)))
129	123, 128	syl5eq 2668	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘0) = (2nd ‘(𝐹‘0)))
130	129	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐵‘0) = (2nd ‘(𝐹‘0)))
131	119, 122, 130	3eqtrd 2660	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (2nd ‘(𝐹‘0)))
132	118, 131	opeq12d 4410	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ⟨(𝑠𝐺1), (𝑠𝐻1)⟩ = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
133		1elunit 12291	. . . . . 6 ⊢ 1 ∈ (0[,]1)
134		oveq12 6659	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑥𝐺𝑦) = (𝑠𝐺1))
135		oveq12 6659	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑥𝐻𝑦) = (𝑠𝐻1))
136	134, 135	opeq12d 4410	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
137		opex 4932	. . . . . . 7 ⊢ ⟨(𝑠𝐺1), (𝑠𝐻1)⟩ ∈ V
138	136, 99, 137	ovmpt2a 6791	. . . . . 6 ⊢ ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)1) = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
139	95, 133, 138	sylancl 694	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)1) = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
140		fvex 6201	. . . . . . . 8 ⊢ (𝐹‘0) ∈ V
141	140	fvconst2 6469	. . . . . . 7 ⊢ (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝐹‘0)})‘𝑠) = (𝐹‘0))
142	141	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐹‘0)})‘𝑠) = (𝐹‘0))
143	19	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘0) ∈ (∪ 𝑅 × ∪ 𝑆))
144		1st2nd2 7205	. . . . . . 7 ⊢ ((𝐹‘0) ∈ (∪ 𝑅 × ∪ 𝑆) → (𝐹‘0) = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
145	143, 144	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘0) = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
146	142, 145	eqtrd 2656	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐹‘0)})‘𝑠) = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
147	132, 139, 146	3eqtr4d 2666	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)1) = (((0[,]1) × {(𝐹‘0)})‘𝑠))
148	27, 35, 37	phtpyi 22783	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0𝐺𝑠) = (𝐴‘0) ∧ (1𝐺𝑠) = (𝐴‘1)))
149	148	simpld 475	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐺𝑠) = (𝐴‘0))
150	149, 117	eqtrd 2656	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐺𝑠) = (1st ‘(𝐹‘0)))
151	52, 59, 61	phtpyi 22783	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0𝐻𝑠) = (𝐵‘0) ∧ (1𝐻𝑠) = (𝐵‘1)))
152	151	simpld 475	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐵‘0))
153	152, 130	eqtrd 2656	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (2nd ‘(𝐹‘0)))
154	150, 153	opeq12d 4410	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ⟨(0𝐺𝑠), (0𝐻𝑠)⟩ = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
155		oveq12 6659	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝑥𝐺𝑦) = (0𝐺𝑠))
156		oveq12 6659	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝑥𝐻𝑦) = (0𝐻𝑠))
157	155, 156	opeq12d 4410	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
158		opex 4932	. . . . . . 7 ⊢ ⟨(0𝐺𝑠), (0𝐻𝑠)⟩ ∈ V
159	157, 99, 158	ovmpt2a 6791	. . . . . 6 ⊢ ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
160	17, 95, 159	sylancr 695	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
161	154, 160, 145	3eqtr4d 2666	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = (𝐹‘0))
162	148	simprd 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐺𝑠) = (𝐴‘1))
163	22	fveq1i 6192	. . . . . . . . . 10 ⊢ (𝐴‘1) = (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘1)
164		fvco3 6275	. . . . . . . . . . 11 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 1 ∈ (0[,]1)) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘1) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)))
165	16, 133, 164	sylancl 694	. . . . . . . . . 10 ⊢ (𝜑 → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘1) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)))
166	163, 165	syl5eq 2668	. . . . . . . . 9 ⊢ (𝜑 → (𝐴‘1) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)))
167		ffvelrn 6357	. . . . . . . . . . 11 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ (∪ 𝑅 × ∪ 𝑆))
168	16, 133, 167	sylancl 694	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘1) ∈ (∪ 𝑅 × ∪ 𝑆))
169		fvres 6207	. . . . . . . . . 10 ⊢ ((𝐹‘1) ∈ (∪ 𝑅 × ∪ 𝑆) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)) = (1st ‘(𝐹‘1)))
170	168, 169	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)) = (1st ‘(𝐹‘1)))
171	166, 170	eqtrd 2656	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘1) = (1st ‘(𝐹‘1)))
172	171	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐴‘1) = (1st ‘(𝐹‘1)))
173	162, 172	eqtrd 2656	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐺𝑠) = (1st ‘(𝐹‘1)))
174	151	simprd 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐵‘1))
175	47	fveq1i 6192	. . . . . . . . . 10 ⊢ (𝐵‘1) = (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘1)
176		fvco3 6275	. . . . . . . . . . 11 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 1 ∈ (0[,]1)) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘1) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)))
177	16, 133, 176	sylancl 694	. . . . . . . . . 10 ⊢ (𝜑 → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘1) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)))
178	175, 177	syl5eq 2668	. . . . . . . . 9 ⊢ (𝜑 → (𝐵‘1) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)))
179		fvres 6207	. . . . . . . . . 10 ⊢ ((𝐹‘1) ∈ (∪ 𝑅 × ∪ 𝑆) → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)) = (2nd ‘(𝐹‘1)))
180	168, 179	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)) = (2nd ‘(𝐹‘1)))
181	178, 180	eqtrd 2656	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘1) = (2nd ‘(𝐹‘1)))
182	181	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐵‘1) = (2nd ‘(𝐹‘1)))
183	174, 182	eqtrd 2656	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (2nd ‘(𝐹‘1)))
184	173, 183	opeq12d 4410	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ⟨(1𝐺𝑠), (1𝐻𝑠)⟩ = ⟨(1st ‘(𝐹‘1)), (2nd ‘(𝐹‘1))⟩)
185		oveq12 6659	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑥𝐺𝑦) = (1𝐺𝑠))
186		oveq12 6659	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑥𝐻𝑦) = (1𝐻𝑠))
187	185, 186	opeq12d 4410	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
188		opex 4932	. . . . . . 7 ⊢ ⟨(1𝐺𝑠), (1𝐻𝑠)⟩ ∈ V
189	187, 99, 188	ovmpt2a 6791	. . . . . 6 ⊢ ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
190	133, 95, 189	sylancr 695	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
191	168	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘1) ∈ (∪ 𝑅 × ∪ 𝑆))
192		1st2nd2 7205	. . . . . 6 ⊢ ((𝐹‘1) ∈ (∪ 𝑅 × ∪ 𝑆) → (𝐹‘1) = ⟨(1st ‘(𝐹‘1)), (2nd ‘(𝐹‘1))⟩)
193	191, 192	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘1) = ⟨(1st ‘(𝐹‘1)), (2nd ‘(𝐹‘1))⟩)
194	184, 190, 193	3eqtr4d 2666	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = (𝐹‘1))
195	1, 21, 69, 105, 147, 161, 194	isphtpy2d 22786	. . 3 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) ∈ (𝐹(PHtpy‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})))
196		ne0i 3921	. . 3 ⊢ ((𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) ∈ (𝐹(PHtpy‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})) → (𝐹(PHtpy‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})) ≠ ∅)
197	195, 196	syl 17	. 2 ⊢ (𝜑 → (𝐹(PHtpy‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})) ≠ ∅)
198		isphtpc 22793	. 2 ⊢ (𝐹( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)}) ↔ (𝐹 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ ((0[,]1) × {(𝐹‘0)}) ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (𝐹(PHtpy‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})) ≠ ∅))
199	1, 21, 197, 198	syl3anbrc 1246	1 ⊢ (𝜑 → 𝐹( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)}))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∅c0 3915  {csn 4177  ⟨cop 4183  ∪ cuni 4436   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112   ↾ cres 5116   ∘ ccom 5118   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1^st c1st 7166  2^nd c2nd 7167  0cc0 9936  1c1 9937  [,]cicc 12178  Topctop 20698  TopOnctopon 20715   Cn ccn 21028   ×_t ctx 21363  IIcii 22678   Htpy chtpy 22766  PHtpycphtpy 22767   ≃_phcphtpc 22768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-cnp 21032  df-tx 21365  df-ii 22680  df-htpy 22769  df-phtpy 22770  df-phtpc 22791

This theorem is referenced by:  txsconn  31223