Theorem txsconnlem 35208

Description: Lemma for txsconn 35209. (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 5762	. . 3 ⊢ ((0[,]1) × {(𝐹‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐹‘0))
3		iitopon 24924	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
4	3	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
5		txsconn.1	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Top)
6		eqid 2740	. . . . . . 7 ⊢ ∪ 𝑅 = ∪ 𝑅
7	6	toptopon 22944	. . . . . 6 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅))
8	5, 7	sylib 218	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (TopOn‘∪ 𝑅))
9		txsconn.2	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Top)
10		eqid 2740	. . . . . . 7 ⊢ ∪ 𝑆 = ∪ 𝑆
11	10	toptopon 22944	. . . . . 6 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘∪ 𝑆))
12	9, 11	sylib 218	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (TopOn‘∪ 𝑆))
13		txtopon 23620	. . . . 5 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
14	8, 12, 13	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝑅 ×t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
15		cnf2 23278	. . . . . 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 1371	. . . . 5 ⊢ (𝜑 → 𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆))
17		0elunit 13529	. . . . 5 ⊢ 0 ∈ (0[,]1)
18		ffvelcdm 7115	. . . . 5 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ (∪ 𝑅 × ∪ 𝑆))
19	16, 17, 18	sylancl 585	. . . 4 ⊢ (𝜑 → (𝐹‘0) ∈ (∪ 𝑅 × ∪ 𝑆))
20	4, 14, 19	cnmptc 23691	. . 3 ⊢ (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐹‘0)) ∈ (II Cn (𝑅 ×t 𝑆)))
21	2, 20	eqeltrid 2848	. 2 ⊢ (𝜑 → ((0[,]1) × {(𝐹‘0)}) ∈ (II Cn (𝑅 ×t 𝑆)))
22		txsconn.5	. . . . . . . . . . 11 ⊢ 𝐴 = ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)
23		tx1cn 23638	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
24	8, 12, 23	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
25		cnco 23295	. . . . . . . . . . . 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 583	. . . . . . . . . . 11 ⊢ (𝜑 → ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑅))
27	22, 26	eqeltrid 2848	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (II Cn 𝑅))
28		fconstmpt 5762	. . . . . . . . . . 11 ⊢ ((0[,]1) × {(𝐴‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐴‘0))
29		iiuni 24926	. . . . . . . . . . . . . . 15 ⊢ (0[,]1) = ∪ II
30	29, 6	cnf 23275	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (II Cn 𝑅) → 𝐴:(0[,]1)⟶∪ 𝑅)
31	27, 30	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴:(0[,]1)⟶∪ 𝑅)
32		ffvelcdm 7115	. . . . . . . . . . . . 13 ⊢ ((𝐴:(0[,]1)⟶∪ 𝑅 ∧ 0 ∈ (0[,]1)) → (𝐴‘0) ∈ ∪ 𝑅)
33	31, 17, 32	sylancl 585	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴‘0) ∈ ∪ 𝑅)
34	4, 8, 33	cnmptc 23691	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐴‘0)) ∈ (II Cn 𝑅))
35	28, 34	eqeltrid 2848	. . . . . . . . . 10 ⊢ (𝜑 → ((0[,]1) × {(𝐴‘0)}) ∈ (II Cn 𝑅))
36	27, 35	phtpycn 25034	. . . . . . . . 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 4009	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝑅))
39		iitop 24925	. . . . . . . . . 10 ⊢ II ∈ Top
40	39, 39, 29, 29	txunii 23622	. . . . . . . . 9 ⊢ ((0[,]1) × (0[,]1)) = ∪ (II ×t II)
41	40, 6	cnf 23275	. . . . . . . 8 ⊢ (𝐺 ∈ ((II ×t II) Cn 𝑅) → 𝐺:((0[,]1) × (0[,]1))⟶∪ 𝑅)
42		ffn 6747	. . . . . . . 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 7581	. . . . . . 7 ⊢ (𝐺 Fn ((0[,]1) × (0[,]1)) ↔ 𝐺 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐺𝑦)))
45	43, 44	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐺𝑦)))
46	45, 38	eqeltrrd 2845	. . . . 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 23639	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
49	8, 12, 48	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → (2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
50		cnco 23295	. . . . . . . . . . . 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 583	. . . . . . . . . . 11 ⊢ (𝜑 → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑆))
52	47, 51	eqeltrid 2848	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ (II Cn 𝑆))
53		fconstmpt 5762	. . . . . . . . . . 11 ⊢ ((0[,]1) × {(𝐵‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐵‘0))
54	29, 10	cnf 23275	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (II Cn 𝑆) → 𝐵:(0[,]1)⟶∪ 𝑆)
55	52, 54	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵:(0[,]1)⟶∪ 𝑆)
56		ffvelcdm 7115	. . . . . . . . . . . . 13 ⊢ ((𝐵:(0[,]1)⟶∪ 𝑆 ∧ 0 ∈ (0[,]1)) → (𝐵‘0) ∈ ∪ 𝑆)
57	55, 17, 56	sylancl 585	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵‘0) ∈ ∪ 𝑆)
58	4, 12, 57	cnmptc 23691	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐵‘0)) ∈ (II Cn 𝑆))
59	53, 58	eqeltrid 2848	. . . . . . . . . 10 ⊢ (𝜑 → ((0[,]1) × {(𝐵‘0)}) ∈ (II Cn 𝑆))
60	52, 59	phtpycn 25034	. . . . . . . . 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 4009	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ ((II ×t II) Cn 𝑆))
63	40, 10	cnf 23275	. . . . . . . 8 ⊢ (𝐻 ∈ ((II ×t II) Cn 𝑆) → 𝐻:((0[,]1) × (0[,]1))⟶∪ 𝑆)
64		ffn 6747	. . . . . . . 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 7581	. . . . . . 7 ⊢ (𝐻 Fn ((0[,]1) × (0[,]1)) ↔ 𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)))
67	65, 66	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)))
68	67, 62	eqeltrrd 2845	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)) ∈ ((II ×t II) Cn 𝑆))
69	4, 4, 46, 68	cnmpt2t 23702	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) ∈ ((II ×t II) Cn (𝑅 ×t 𝑆)))
70	27, 35	phtpyhtpy 25033	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)})) ⊆ (𝐴(II Htpy 𝑅)((0[,]1) × {(𝐴‘0)})))
71	70, 37	sseldd 4009	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (𝐴(II Htpy 𝑅)((0[,]1) × {(𝐴‘0)})))
72	4, 27, 35, 71	htpyi 25025	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝑠𝐺0) = (𝐴‘𝑠) ∧ (𝑠𝐺1) = (((0[,]1) × {(𝐴‘0)})‘𝑠)))
73	72	simpld 494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐺0) = (𝐴‘𝑠))
74	22	fveq1i 6921	. . . . . . . 8 ⊢ (𝐴‘𝑠) = (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘𝑠)
75		fvco3 7021	. . . . . . . . 9 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘𝑠) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)))
76	16, 75	sylan 579	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘𝑠) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)))
77	74, 76	eqtrid 2792	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐴‘𝑠) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)))
78		ffvelcdm 7115	. . . . . . . . 9 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘𝑠) ∈ (∪ 𝑅 × ∪ 𝑆))
79	16, 78	sylan 579	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘𝑠) ∈ (∪ 𝑅 × ∪ 𝑆))
80		fvres 6939	. . . . . . . 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 2784	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐺0) = (1st ‘(𝐹‘𝑠)))
83	52, 59	phtpyhtpy 25033	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)})) ⊆ (𝐵(II Htpy 𝑆)((0[,]1) × {(𝐵‘0)})))
84	83, 61	sseldd 4009	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ (𝐵(II Htpy 𝑆)((0[,]1) × {(𝐵‘0)})))
85	4, 52, 59, 84	htpyi 25025	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝑠𝐻0) = (𝐵‘𝑠) ∧ (𝑠𝐻1) = (((0[,]1) × {(𝐵‘0)})‘𝑠)))
86	85	simpld 494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐵‘𝑠))
87	47	fveq1i 6921	. . . . . . . 8 ⊢ (𝐵‘𝑠) = (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘𝑠)
88		fvco3 7021	. . . . . . . . 9 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘𝑠) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)))
89	16, 88	sylan 579	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘𝑠) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)))
90	87, 89	eqtrid 2792	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐵‘𝑠) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)))
91		fvres 6939	. . . . . . . 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 2784	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (2nd ‘(𝐹‘𝑠)))
94	82, 93	opeq12d 4905	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ⟨(𝑠𝐺0), (𝑠𝐻0)⟩ = ⟨(1st ‘(𝐹‘𝑠)), (2nd ‘(𝐹‘𝑠))⟩)
95		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ (0[,]1))
96		oveq12 7457	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑥𝐺𝑦) = (𝑠𝐺0))
97		oveq12 7457	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑥𝐻𝑦) = (𝑠𝐻0))
98	96, 97	opeq12d 4905	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
99		eqid 2740	. . . . . . 7 ⊢ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)
100		opex 5484	. . . . . . 7 ⊢ ⟨(𝑠𝐺0), (𝑠𝐻0)⟩ ∈ V
101	98, 99, 100	ovmpoa 7605	. . . . . 6 ⊢ ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
102	95, 17, 101	sylancl 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
103		1st2nd2 8069	. . . . . 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 2790	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = (𝐹‘𝑠))
106	72	simprd 495	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐺1) = (((0[,]1) × {(𝐴‘0)})‘𝑠))
107		fvex 6933	. . . . . . . . 9 ⊢ (𝐴‘0) ∈ V
108	107	fvconst2 7241	. . . . . . . 8 ⊢ (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝐴‘0)})‘𝑠) = (𝐴‘0))
109	108	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐴‘0)})‘𝑠) = (𝐴‘0))
110	22	fveq1i 6921	. . . . . . . . 9 ⊢ (𝐴‘0) = (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0)
111		fvco3 7021	. . . . . . . . . . 11 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 0 ∈ (0[,]1)) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)))
112	16, 17, 111	sylancl 585	. . . . . . . . . 10 ⊢ (𝜑 → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)))
113		fvres 6939	. . . . . . . . . . 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 2780	. . . . . . . . 9 ⊢ (𝜑 → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0) = (1st ‘(𝐹‘0)))
116	110, 115	eqtrid 2792	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘0) = (1st ‘(𝐹‘0)))
117	116	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐴‘0) = (1st ‘(𝐹‘0)))
118	106, 109, 117	3eqtrd 2784	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐺1) = (1st ‘(𝐹‘0)))
119	85	simprd 495	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (((0[,]1) × {(𝐵‘0)})‘𝑠))
120		fvex 6933	. . . . . . . . 9 ⊢ (𝐵‘0) ∈ V
121	120	fvconst2 7241	. . . . . . . 8 ⊢ (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝐵‘0)})‘𝑠) = (𝐵‘0))
122	121	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐵‘0)})‘𝑠) = (𝐵‘0))
123	47	fveq1i 6921	. . . . . . . . 9 ⊢ (𝐵‘0) = (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0)
124		fvco3 7021	. . . . . . . . . . 11 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 0 ∈ (0[,]1)) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)))
125	16, 17, 124	sylancl 585	. . . . . . . . . 10 ⊢ (𝜑 → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)))
126		fvres 6939	. . . . . . . . . . 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 2780	. . . . . . . . 9 ⊢ (𝜑 → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0) = (2nd ‘(𝐹‘0)))
129	123, 128	eqtrid 2792	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘0) = (2nd ‘(𝐹‘0)))
130	129	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐵‘0) = (2nd ‘(𝐹‘0)))
131	119, 122, 130	3eqtrd 2784	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (2nd ‘(𝐹‘0)))
132	118, 131	opeq12d 4905	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ⟨(𝑠𝐺1), (𝑠𝐻1)⟩ = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
133		1elunit 13530	. . . . . 6 ⊢ 1 ∈ (0[,]1)
134		oveq12 7457	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑥𝐺𝑦) = (𝑠𝐺1))
135		oveq12 7457	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑥𝐻𝑦) = (𝑠𝐻1))
136	134, 135	opeq12d 4905	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
137		opex 5484	. . . . . . 7 ⊢ ⟨(𝑠𝐺1), (𝑠𝐻1)⟩ ∈ V
138	136, 99, 137	ovmpoa 7605	. . . . . 6 ⊢ ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)1) = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
139	95, 133, 138	sylancl 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)1) = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
140		fvex 6933	. . . . . . . 8 ⊢ (𝐹‘0) ∈ V
141	140	fvconst2 7241	. . . . . . 7 ⊢ (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝐹‘0)})‘𝑠) = (𝐹‘0))
142	141	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐹‘0)})‘𝑠) = (𝐹‘0))
143	19	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘0) ∈ (∪ 𝑅 × ∪ 𝑆))
144		1st2nd2 8069	. . . . . . 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 2780	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐹‘0)})‘𝑠) = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
147	132, 139, 146	3eqtr4d 2790	. . . 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 25035	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0𝐺𝑠) = (𝐴‘0) ∧ (1𝐺𝑠) = (𝐴‘1)))
149	148	simpld 494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐺𝑠) = (𝐴‘0))
150	149, 117	eqtrd 2780	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐺𝑠) = (1st ‘(𝐹‘0)))
151	52, 59, 61	phtpyi 25035	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0𝐻𝑠) = (𝐵‘0) ∧ (1𝐻𝑠) = (𝐵‘1)))
152	151	simpld 494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐵‘0))
153	152, 130	eqtrd 2780	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (2nd ‘(𝐹‘0)))
154	150, 153	opeq12d 4905	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ⟨(0𝐺𝑠), (0𝐻𝑠)⟩ = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
155		oveq12 7457	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝑥𝐺𝑦) = (0𝐺𝑠))
156		oveq12 7457	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝑥𝐻𝑦) = (0𝐻𝑠))
157	155, 156	opeq12d 4905	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
158		opex 5484	. . . . . . 7 ⊢ ⟨(0𝐺𝑠), (0𝐻𝑠)⟩ ∈ V
159	157, 99, 158	ovmpoa 7605	. . . . . 6 ⊢ ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
160	17, 95, 159	sylancr 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
161	154, 160, 145	3eqtr4d 2790	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = (𝐹‘0))
162	148	simprd 495	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐺𝑠) = (𝐴‘1))
163	22	fveq1i 6921	. . . . . . . . . 10 ⊢ (𝐴‘1) = (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘1)
164		fvco3 7021	. . . . . . . . . . 11 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 1 ∈ (0[,]1)) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘1) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)))
165	16, 133, 164	sylancl 585	. . . . . . . . . 10 ⊢ (𝜑 → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘1) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)))
166	163, 165	eqtrid 2792	. . . . . . . . 9 ⊢ (𝜑 → (𝐴‘1) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)))
167		ffvelcdm 7115	. . . . . . . . . . 11 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ (∪ 𝑅 × ∪ 𝑆))
168	16, 133, 167	sylancl 585	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘1) ∈ (∪ 𝑅 × ∪ 𝑆))
169		fvres 6939	. . . . . . . . . 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 2780	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘1) = (1st ‘(𝐹‘1)))
172	171	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐴‘1) = (1st ‘(𝐹‘1)))
173	162, 172	eqtrd 2780	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐺𝑠) = (1st ‘(𝐹‘1)))
174	151	simprd 495	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐵‘1))
175	47	fveq1i 6921	. . . . . . . . . 10 ⊢ (𝐵‘1) = (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘1)
176		fvco3 7021	. . . . . . . . . . 11 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 1 ∈ (0[,]1)) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘1) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)))
177	16, 133, 176	sylancl 585	. . . . . . . . . 10 ⊢ (𝜑 → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘1) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)))
178	175, 177	eqtrid 2792	. . . . . . . . 9 ⊢ (𝜑 → (𝐵‘1) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)))
179		fvres 6939	. . . . . . . . . 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 2780	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘1) = (2nd ‘(𝐹‘1)))
182	181	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐵‘1) = (2nd ‘(𝐹‘1)))
183	174, 182	eqtrd 2780	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (2nd ‘(𝐹‘1)))
184	173, 183	opeq12d 4905	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ⟨(1𝐺𝑠), (1𝐻𝑠)⟩ = ⟨(1st ‘(𝐹‘1)), (2nd ‘(𝐹‘1))⟩)
185		oveq12 7457	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑥𝐺𝑦) = (1𝐺𝑠))
186		oveq12 7457	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑥𝐻𝑦) = (1𝐻𝑠))
187	185, 186	opeq12d 4905	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
188		opex 5484	. . . . . . 7 ⊢ ⟨(1𝐺𝑠), (1𝐻𝑠)⟩ ∈ V
189	187, 99, 188	ovmpoa 7605	. . . . . 6 ⊢ ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
190	133, 95, 189	sylancr 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
191	168	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘1) ∈ (∪ 𝑅 × ∪ 𝑆))
192		1st2nd2 8069	. . . . . 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 2790	. . . 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 25038	. . 3 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) ∈ (𝐹(PHtpy‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})))
196	195	ne0d 4365	. 2 ⊢ (𝜑 → (𝐹(PHtpy‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})) ≠ ∅)
197		isphtpc 25045	. 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)})) ≠ ∅))
198	1, 21, 196, 197	syl3anbrc 1343	1 ⊢ (𝜑 → 𝐹( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∅c0 4352  {csn 4648  ⟨cop 4654  ∪ cuni 4931   class class class wbr 5166   ↦ cmpt 5249   × cxp 5698   ↾ cres 5702   ∘ ccom 5704   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  1^st c1st 8028  2^nd c2nd 8029  0cc0 11184  1c1 11185  [,]cicc 13410  Topctop 22920  TopOnctopon 22937   Cn ccn 23253   ×_t ctx 23589  IIcii 24920   Htpy chtpy 25018  PHtpycphtpy 25019   ≃_phcphtpc 25020

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-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-iun 5017  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-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-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-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-sup 9511  df-inf 9512  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-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-tx 23591  df-ii 24922  df-htpy 25021  df-phtpy 25022  df-phtpc 25043

This theorem is referenced by:  txsconn  35209