Theorem txsconnlem 36005

Description: Lemma for txsconn 36006. (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 5712	. . 3 ⊢ ((0[,]1) × {(𝐹‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐹‘0))
3		iitopon 25200	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
4	3	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
5		txsconn.1	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Top)
6		eqid 2760	. . . . . . 7 ⊢ ∪ 𝑅 = ∪ 𝑅
7	6	toptopon 23235	. . . . . 6 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘∪ 𝑅))
8	5, 7	sylib 221	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (TopOn‘∪ 𝑅))
9		txsconn.2	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Top)
10		eqid 2760	. . . . . . 7 ⊢ ∪ 𝑆 = ∪ 𝑆
11	10	toptopon 23235	. . . . . 6 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘∪ 𝑆))
12	9, 11	sylib 221	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (TopOn‘∪ 𝑆))
13		txtopon 23910	. . . . 5 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
14	8, 12, 13	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝑅 ×t 𝑆) ∈ (TopOn‘(∪ 𝑅 × ∪ 𝑆)))
15		cnf2 23567	. . . . . 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 1398	. . . . 5 ⊢ (𝜑 → 𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆))
17		0elunit 13600	. . . . 5 ⊢ 0 ∈ (0[,]1)
18		ffvelcdm 7081	. . . . 5 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ (∪ 𝑅 × ∪ 𝑆))
19	16, 17, 18	sylancl 598	. . . 4 ⊢ (𝜑 → (𝐹‘0) ∈ (∪ 𝑅 × ∪ 𝑆))
20	4, 14, 19	cnmptc 23981	. . 3 ⊢ (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐹‘0)) ∈ (II Cn (𝑅 ×t 𝑆)))
21	2, 20	eqeltrid 2864	. 2 ⊢ (𝜑 → ((0[,]1) × {(𝐹‘0)}) ∈ (II Cn (𝑅 ×t 𝑆)))
22		txsconn.5	. . . . . . . . . . 11 ⊢ 𝐴 = ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)
23		tx1cn 23928	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
24	8, 12, 23	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
25		cnco 23584	. . . . . . . . . . . 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 596	. . . . . . . . . . 11 ⊢ (𝜑 → ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑅))
27	22, 26	eqeltrid 2864	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (II Cn 𝑅))
28		fconstmpt 5712	. . . . . . . . . . 11 ⊢ ((0[,]1) × {(𝐴‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐴‘0))
29		iiuni 25202	. . . . . . . . . . . . . . 15 ⊢ (0[,]1) = ∪ II
30	29, 6	cnf 23564	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (II Cn 𝑅) → 𝐴:(0[,]1)⟶∪ 𝑅)
31	27, 30	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴:(0[,]1)⟶∪ 𝑅)
32		ffvelcdm 7081	. . . . . . . . . . . . 13 ⊢ ((𝐴:(0[,]1)⟶∪ 𝑅 ∧ 0 ∈ (0[,]1)) → (𝐴‘0) ∈ ∪ 𝑅)
33	31, 17, 32	sylancl 598	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴‘0) ∈ ∪ 𝑅)
34	4, 8, 33	cnmptc 23981	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐴‘0)) ∈ (II Cn 𝑅))
35	28, 34	eqeltrid 2864	. . . . . . . . . 10 ⊢ (𝜑 → ((0[,]1) × {(𝐴‘0)}) ∈ (II Cn 𝑅))
36	27, 35	phtpycn 25304	. . . . . . . . 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 3909	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝑅))
39		iitop 25201	. . . . . . . . . 10 ⊢ II ∈ Top
40	39, 39, 29, 29	txunii 23912	. . . . . . . . 9 ⊢ ((0[,]1) × (0[,]1)) = ∪ (II ×t II)
41	40, 6	cnf 23564	. . . . . . . 8 ⊢ (𝐺 ∈ ((II ×t II) Cn 𝑅) → 𝐺:((0[,]1) × (0[,]1))⟶∪ 𝑅)
42		ffn 6709	. . . . . . . 8 ⊢ (𝐺:((0[,]1) × (0[,]1))⟶∪ 𝑅 → 𝐺 Fn ((0[,]1) × (0[,]1)))
43	38, 41, 42	3syl 19	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ((0[,]1) × (0[,]1)))
44		fnov 7551	. . . . . . 7 ⊢ (𝐺 Fn ((0[,]1) × (0[,]1)) ↔ 𝐺 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐺𝑦)))
45	43, 44	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐺𝑦)))
46	45, 38	eqeltrrd 2861	. . . . 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 23929	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ (TopOn‘∪ 𝑅) ∧ 𝑆 ∈ (TopOn‘∪ 𝑆)) → (2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
49	8, 12, 48	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
50		cnco 23584	. . . . . . . . . . . 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 596	. . . . . . . . . . 11 ⊢ (𝜑 → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑆))
52	47, 51	eqeltrid 2864	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ (II Cn 𝑆))
53		fconstmpt 5712	. . . . . . . . . . 11 ⊢ ((0[,]1) × {(𝐵‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐵‘0))
54	29, 10	cnf 23564	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (II Cn 𝑆) → 𝐵:(0[,]1)⟶∪ 𝑆)
55	52, 54	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵:(0[,]1)⟶∪ 𝑆)
56		ffvelcdm 7081	. . . . . . . . . . . . 13 ⊢ ((𝐵:(0[,]1)⟶∪ 𝑆 ∧ 0 ∈ (0[,]1)) → (𝐵‘0) ∈ ∪ 𝑆)
57	55, 17, 56	sylancl 598	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵‘0) ∈ ∪ 𝑆)
58	4, 12, 57	cnmptc 23981	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐵‘0)) ∈ (II Cn 𝑆))
59	53, 58	eqeltrid 2864	. . . . . . . . . 10 ⊢ (𝜑 → ((0[,]1) × {(𝐵‘0)}) ∈ (II Cn 𝑆))
60	52, 59	phtpycn 25304	. . . . . . . . 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 3909	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ ((II ×t II) Cn 𝑆))
63	40, 10	cnf 23564	. . . . . . . 8 ⊢ (𝐻 ∈ ((II ×t II) Cn 𝑆) → 𝐻:((0[,]1) × (0[,]1))⟶∪ 𝑆)
64		ffn 6709	. . . . . . . 8 ⊢ (𝐻:((0[,]1) × (0[,]1))⟶∪ 𝑆 → 𝐻 Fn ((0[,]1) × (0[,]1)))
65	62, 63, 64	3syl 19	. . . . . . 7 ⊢ (𝜑 → 𝐻 Fn ((0[,]1) × (0[,]1)))
66		fnov 7551	. . . . . . 7 ⊢ (𝐻 Fn ((0[,]1) × (0[,]1)) ↔ 𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)))
67	65, 66	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)))
68	67, 62	eqeltrrd 2861	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)) ∈ ((II ×t II) Cn 𝑆))
69	4, 4, 46, 68	cnmpt2t 23992	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) ∈ ((II ×t II) Cn (𝑅 ×t 𝑆)))
70	27, 35	phtpyhtpy 25303	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)})) ⊆ (𝐴(II Htpy 𝑅)((0[,]1) × {(𝐴‘0)})))
71	70, 37	sseldd 3909	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (𝐴(II Htpy 𝑅)((0[,]1) × {(𝐴‘0)})))
72	4, 27, 35, 71	htpyi 25295	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝑠𝐺0) = (𝐴‘𝑠) ∧ (𝑠𝐺1) = (((0[,]1) × {(𝐴‘0)})‘𝑠)))
73	72	simpld 500	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐺0) = (𝐴‘𝑠))
74	22	fveq1i 6886	. . . . . . . 8 ⊢ (𝐴‘𝑠) = (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘𝑠)
75		fvco3 6985	. . . . . . . . 9 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘𝑠) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)))
76	16, 75	sylan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘𝑠) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)))
77	74, 76	eqtrid 2807	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐴‘𝑠) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)))
78		ffvelcdm 7081	. . . . . . . . 9 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘𝑠) ∈ (∪ 𝑅 × ∪ 𝑆))
79	16, 78	sylan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘𝑠) ∈ (∪ 𝑅 × ∪ 𝑆))
80		fvres 6904	. . . . . . . 8 ⊢ ((𝐹‘𝑠) ∈ (∪ 𝑅 × ∪ 𝑆) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)) = (1st ‘(𝐹‘𝑠)))
81	79, 80	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)) = (1st ‘(𝐹‘𝑠)))
82	73, 77, 81	3eqtrd 2799	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐺0) = (1st ‘(𝐹‘𝑠)))
83	52, 59	phtpyhtpy 25303	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)})) ⊆ (𝐵(II Htpy 𝑆)((0[,]1) × {(𝐵‘0)})))
84	83, 61	sseldd 3909	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ (𝐵(II Htpy 𝑆)((0[,]1) × {(𝐵‘0)})))
85	4, 52, 59, 84	htpyi 25295	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝑠𝐻0) = (𝐵‘𝑠) ∧ (𝑠𝐻1) = (((0[,]1) × {(𝐵‘0)})‘𝑠)))
86	85	simpld 500	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐵‘𝑠))
87	47	fveq1i 6886	. . . . . . . 8 ⊢ (𝐵‘𝑠) = (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘𝑠)
88		fvco3 6985	. . . . . . . . 9 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘𝑠) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)))
89	16, 88	sylan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘𝑠) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)))
90	87, 89	eqtrid 2807	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐵‘𝑠) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)))
91		fvres 6904	. . . . . . . 8 ⊢ ((𝐹‘𝑠) ∈ (∪ 𝑅 × ∪ 𝑆) → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)) = (2nd ‘(𝐹‘𝑠)))
92	79, 91	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘𝑠)) = (2nd ‘(𝐹‘𝑠)))
93	86, 90, 92	3eqtrd 2799	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (2nd ‘(𝐹‘𝑠)))
94	82, 93	opeq12d 4840	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ⟨(𝑠𝐺0), (𝑠𝐻0)⟩ = ⟨(1st ‘(𝐹‘𝑠)), (2nd ‘(𝐹‘𝑠))⟩)
95		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ (0[,]1))
96		oveq12 7429	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑥𝐺𝑦) = (𝑠𝐺0))
97		oveq12 7429	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑥𝐻𝑦) = (𝑠𝐻0))
98	96, 97	opeq12d 4840	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
99		eqid 2760	. . . . . . 7 ⊢ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)
100		opex 5431	. . . . . . 7 ⊢ ⟨(𝑠𝐺0), (𝑠𝐻0)⟩ ∈ V
101	98, 99, 100	ovmpoa 7575	. . . . . 6 ⊢ ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
102	95, 17, 101	sylancl 598	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
103		1st2nd2 8040	. . . . . 6 ⊢ ((𝐹‘𝑠) ∈ (∪ 𝑅 × ∪ 𝑆) → (𝐹‘𝑠) = ⟨(1st ‘(𝐹‘𝑠)), (2nd ‘(𝐹‘𝑠))⟩)
104	79, 103	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘𝑠) = ⟨(1st ‘(𝐹‘𝑠)), (2nd ‘(𝐹‘𝑠))⟩)
105	94, 102, 104	3eqtr4d 2805	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = (𝐹‘𝑠))
106	72	simprd 501	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐺1) = (((0[,]1) × {(𝐴‘0)})‘𝑠))
107		fvex 6898	. . . . . . . . 9 ⊢ (𝐴‘0) ∈ V
108	107	fvconst2 7210	. . . . . . . 8 ⊢ (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝐴‘0)})‘𝑠) = (𝐴‘0))
109	108	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐴‘0)})‘𝑠) = (𝐴‘0))
110	22	fveq1i 6886	. . . . . . . . 9 ⊢ (𝐴‘0) = (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0)
111		fvco3 6985	. . . . . . . . . . 11 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 0 ∈ (0[,]1)) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)))
112	16, 17, 111	sylancl 598	. . . . . . . . . 10 ⊢ (𝜑 → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)))
113		fvres 6904	. . . . . . . . . . 11 ⊢ ((𝐹‘0) ∈ (∪ 𝑅 × ∪ 𝑆) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)) = (1st ‘(𝐹‘0)))
114	19, 113	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)) = (1st ‘(𝐹‘0)))
115	112, 114	eqtrd 2795	. . . . . . . . 9 ⊢ (𝜑 → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0) = (1st ‘(𝐹‘0)))
116	110, 115	eqtrid 2807	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘0) = (1st ‘(𝐹‘0)))
117	116	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐴‘0) = (1st ‘(𝐹‘0)))
118	106, 109, 117	3eqtrd 2799	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐺1) = (1st ‘(𝐹‘0)))
119	85	simprd 501	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (((0[,]1) × {(𝐵‘0)})‘𝑠))
120		fvex 6898	. . . . . . . . 9 ⊢ (𝐵‘0) ∈ V
121	120	fvconst2 7210	. . . . . . . 8 ⊢ (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝐵‘0)})‘𝑠) = (𝐵‘0))
122	121	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐵‘0)})‘𝑠) = (𝐵‘0))
123	47	fveq1i 6886	. . . . . . . . 9 ⊢ (𝐵‘0) = (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0)
124		fvco3 6985	. . . . . . . . . . 11 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 0 ∈ (0[,]1)) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)))
125	16, 17, 124	sylancl 598	. . . . . . . . . 10 ⊢ (𝜑 → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)))
126		fvres 6904	. . . . . . . . . . 11 ⊢ ((𝐹‘0) ∈ (∪ 𝑅 × ∪ 𝑆) → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)) = (2nd ‘(𝐹‘0)))
127	19, 126	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘0)) = (2nd ‘(𝐹‘0)))
128	125, 127	eqtrd 2795	. . . . . . . . 9 ⊢ (𝜑 → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘0) = (2nd ‘(𝐹‘0)))
129	123, 128	eqtrid 2807	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘0) = (2nd ‘(𝐹‘0)))
130	129	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐵‘0) = (2nd ‘(𝐹‘0)))
131	119, 122, 130	3eqtrd 2799	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (2nd ‘(𝐹‘0)))
132	118, 131	opeq12d 4840	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ⟨(𝑠𝐺1), (𝑠𝐻1)⟩ = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
133		1elunit 13601	. . . . . 6 ⊢ 1 ∈ (0[,]1)
134		oveq12 7429	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑥𝐺𝑦) = (𝑠𝐺1))
135		oveq12 7429	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑥𝐻𝑦) = (𝑠𝐻1))
136	134, 135	opeq12d 4840	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
137		opex 5431	. . . . . . 7 ⊢ ⟨(𝑠𝐺1), (𝑠𝐻1)⟩ ∈ V
138	136, 99, 137	ovmpoa 7575	. . . . . 6 ⊢ ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)1) = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
139	95, 133, 138	sylancl 598	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)1) = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
140		fvex 6898	. . . . . . . 8 ⊢ (𝐹‘0) ∈ V
141	140	fvconst2 7210	. . . . . . 7 ⊢ (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝐹‘0)})‘𝑠) = (𝐹‘0))
142	141	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐹‘0)})‘𝑠) = (𝐹‘0))
143	19	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘0) ∈ (∪ 𝑅 × ∪ 𝑆))
144		1st2nd2 8040	. . . . . . 7 ⊢ ((𝐹‘0) ∈ (∪ 𝑅 × ∪ 𝑆) → (𝐹‘0) = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
145	143, 144	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘0) = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
146	142, 145	eqtrd 2795	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐹‘0)})‘𝑠) = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
147	132, 139, 146	3eqtr4d 2805	. . . 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 25305	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0𝐺𝑠) = (𝐴‘0) ∧ (1𝐺𝑠) = (𝐴‘1)))
149	148	simpld 500	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐺𝑠) = (𝐴‘0))
150	149, 117	eqtrd 2795	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐺𝑠) = (1st ‘(𝐹‘0)))
151	52, 59, 61	phtpyi 25305	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0𝐻𝑠) = (𝐵‘0) ∧ (1𝐻𝑠) = (𝐵‘1)))
152	151	simpld 500	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐵‘0))
153	152, 130	eqtrd 2795	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (2nd ‘(𝐹‘0)))
154	150, 153	opeq12d 4840	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ⟨(0𝐺𝑠), (0𝐻𝑠)⟩ = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
155		oveq12 7429	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝑥𝐺𝑦) = (0𝐺𝑠))
156		oveq12 7429	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝑥𝐻𝑦) = (0𝐻𝑠))
157	155, 156	opeq12d 4840	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
158		opex 5431	. . . . . . 7 ⊢ ⟨(0𝐺𝑠), (0𝐻𝑠)⟩ ∈ V
159	157, 99, 158	ovmpoa 7575	. . . . . 6 ⊢ ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
160	17, 95, 159	sylancr 599	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
161	154, 160, 145	3eqtr4d 2805	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = (𝐹‘0))
162	148	simprd 501	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐺𝑠) = (𝐴‘1))
163	22	fveq1i 6886	. . . . . . . . . 10 ⊢ (𝐴‘1) = (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘1)
164		fvco3 6985	. . . . . . . . . . 11 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 1 ∈ (0[,]1)) → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘1) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)))
165	16, 133, 164	sylancl 598	. . . . . . . . . 10 ⊢ (𝜑 → (((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘1) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)))
166	163, 165	eqtrid 2807	. . . . . . . . 9 ⊢ (𝜑 → (𝐴‘1) = ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)))
167		ffvelcdm 7081	. . . . . . . . . . 11 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ (∪ 𝑅 × ∪ 𝑆))
168	16, 133, 167	sylancl 598	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘1) ∈ (∪ 𝑅 × ∪ 𝑆))
169		fvres 6904	. . . . . . . . . 10 ⊢ ((𝐹‘1) ∈ (∪ 𝑅 × ∪ 𝑆) → ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)) = (1st ‘(𝐹‘1)))
170	168, 169	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((1st ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)) = (1st ‘(𝐹‘1)))
171	166, 170	eqtrd 2795	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘1) = (1st ‘(𝐹‘1)))
172	171	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐴‘1) = (1st ‘(𝐹‘1)))
173	162, 172	eqtrd 2795	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐺𝑠) = (1st ‘(𝐹‘1)))
174	151	simprd 501	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐵‘1))
175	47	fveq1i 6886	. . . . . . . . . 10 ⊢ (𝐵‘1) = (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘1)
176		fvco3 6985	. . . . . . . . . . 11 ⊢ ((𝐹:(0[,]1)⟶(∪ 𝑅 × ∪ 𝑆) ∧ 1 ∈ (0[,]1)) → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘1) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)))
177	16, 133, 176	sylancl 598	. . . . . . . . . 10 ⊢ (𝜑 → (((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹)‘1) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)))
178	175, 177	eqtrid 2807	. . . . . . . . 9 ⊢ (𝜑 → (𝐵‘1) = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)))
179		fvres 6904	. . . . . . . . . 10 ⊢ ((𝐹‘1) ∈ (∪ 𝑅 × ∪ 𝑆) → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)) = (2nd ‘(𝐹‘1)))
180	168, 179	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((2nd ↾ (∪ 𝑅 × ∪ 𝑆))‘(𝐹‘1)) = (2nd ‘(𝐹‘1)))
181	178, 180	eqtrd 2795	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘1) = (2nd ‘(𝐹‘1)))
182	181	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐵‘1) = (2nd ‘(𝐹‘1)))
183	174, 182	eqtrd 2795	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (2nd ‘(𝐹‘1)))
184	173, 183	opeq12d 4840	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ⟨(1𝐺𝑠), (1𝐻𝑠)⟩ = ⟨(1st ‘(𝐹‘1)), (2nd ‘(𝐹‘1))⟩)
185		oveq12 7429	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑥𝐺𝑦) = (1𝐺𝑠))
186		oveq12 7429	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑥𝐻𝑦) = (1𝐻𝑠))
187	185, 186	opeq12d 4840	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
188		opex 5431	. . . . . . 7 ⊢ ⟨(1𝐺𝑠), (1𝐻𝑠)⟩ ∈ V
189	187, 99, 188	ovmpoa 7575	. . . . . 6 ⊢ ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
190	133, 95, 189	sylancr 599	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
191	168	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘1) ∈ (∪ 𝑅 × ∪ 𝑆))
192		1st2nd2 8040	. . . . . 6 ⊢ ((𝐹‘1) ∈ (∪ 𝑅 × ∪ 𝑆) → (𝐹‘1) = ⟨(1st ‘(𝐹‘1)), (2nd ‘(𝐹‘1))⟩)
193	191, 192	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘1) = ⟨(1st ‘(𝐹‘1)), (2nd ‘(𝐹‘1))⟩)
194	184, 190, 193	3eqtr4d 2805	. . . 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 25308	. . 3 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) ∈ (𝐹(PHtpy‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})))
196	195	ne0d 4287	. 2 ⊢ (𝜑 → (𝐹(PHtpy‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})) ≠ ∅)
197		isphtpc 25315	. 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 1362	1 ⊢ (𝜑 → 𝐹( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∅c0 4278  {csn 4583  ⟨cop 4589  ∪ cuni 4866   class class class wbr 5102   ↦ cmpt 5185   × cxp 5648   ↾ cres 5652   ∘ ccom 5654   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000  0cc0 11200  1c1 11201  [,]cicc 13479  Topctop 23211  TopOnctopon 23228   Cn ccn 23542   ×_t ctx 23879  IIcii 25196   Htpy chtpy 25288  PHtpycphtpy 25289   ≃_phcphtpc 25290

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-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-iun 4952  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-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-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-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-sup 9434  df-inf 9435  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-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-tx 23881  df-ii 25198  df-htpy 25291  df-phtpy 25292  df-phtpc 25313

This theorem is used by:  txsconn  36006