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Theorem txsconnlem 31196
Description: Lemma for txsconn 31197. (Contributed by Mario Carneiro, 9-Mar-2015.)
Hypotheses
Ref Expression
txsconn.1 (𝜑𝑅 ∈ Top)
txsconn.2 (𝜑𝑆 ∈ Top)
txsconn.3 (𝜑𝐹 ∈ (II Cn (𝑅 ×t 𝑆)))
txsconn.5 𝐴 = ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)
txsconn.6 𝐵 = ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)
txsconn.7 (𝜑𝐺 ∈ (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)})))
txsconn.8 (𝜑𝐻 ∈ (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)})))
Assertion
Ref Expression
txsconnlem (𝜑𝐹( ≃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.
StepHypRef Expression
1 txsconn.3 . 2 (𝜑𝐹 ∈ (II Cn (𝑅 ×t 𝑆)))
2 fconstmpt 5153 . . 3 ((0[,]1) × {(𝐹‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐹‘0))
3 iitopon 22663 . . . . 5 II ∈ (TopOn‘(0[,]1))
43a1i 11 . . . 4 (𝜑 → II ∈ (TopOn‘(0[,]1)))
5 txsconn.1 . . . . . 6 (𝜑𝑅 ∈ Top)
6 eqid 2620 . . . . . . 7 𝑅 = 𝑅
76toptopon 20703 . . . . . 6 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘ 𝑅))
85, 7sylib 208 . . . . 5 (𝜑𝑅 ∈ (TopOn‘ 𝑅))
9 txsconn.2 . . . . . 6 (𝜑𝑆 ∈ Top)
10 eqid 2620 . . . . . . 7 𝑆 = 𝑆
1110toptopon 20703 . . . . . 6 (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘ 𝑆))
129, 11sylib 208 . . . . 5 (𝜑𝑆 ∈ (TopOn‘ 𝑆))
13 txtopon 21375 . . . . 5 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (𝑅 ×t 𝑆) ∈ (TopOn‘( 𝑅 × 𝑆)))
148, 12, 13syl2anc 692 . . . 4 (𝜑 → (𝑅 ×t 𝑆) ∈ (TopOn‘( 𝑅 × 𝑆)))
15 cnf2 21034 . . . . . 6 ((II ∈ (TopOn‘(0[,]1)) ∧ (𝑅 ×t 𝑆) ∈ (TopOn‘( 𝑅 × 𝑆)) ∧ 𝐹 ∈ (II Cn (𝑅 ×t 𝑆))) → 𝐹:(0[,]1)⟶( 𝑅 × 𝑆))
164, 14, 1, 15syl3anc 1324 . . . . 5 (𝜑𝐹:(0[,]1)⟶( 𝑅 × 𝑆))
17 0elunit 12275 . . . . 5 0 ∈ (0[,]1)
18 ffvelrn 6343 . . . . 5 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ ( 𝑅 × 𝑆))
1916, 17, 18sylancl 693 . . . 4 (𝜑 → (𝐹‘0) ∈ ( 𝑅 × 𝑆))
204, 14, 19cnmptc 21446 . . 3 (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐹‘0)) ∈ (II Cn (𝑅 ×t 𝑆)))
212, 20syl5eqel 2703 . 2 (𝜑 → ((0[,]1) × {(𝐹‘0)}) ∈ (II Cn (𝑅 ×t 𝑆)))
22 txsconn.5 . . . . . . . . . . 11 𝐴 = ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)
23 tx1cn 21393 . . . . . . . . . . . . 13 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (1st ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
248, 12, 23syl2anc 692 . . . . . . . . . . . 12 (𝜑 → (1st ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
25 cnco 21051 . . . . . . . . . . . 12 ((𝐹 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (1st ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅)) → ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑅))
261, 24, 25syl2anc 692 . . . . . . . . . . 11 (𝜑 → ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑅))
2722, 26syl5eqel 2703 . . . . . . . . . 10 (𝜑𝐴 ∈ (II Cn 𝑅))
28 fconstmpt 5153 . . . . . . . . . . 11 ((0[,]1) × {(𝐴‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐴‘0))
29 iiuni 22665 . . . . . . . . . . . . . . 15 (0[,]1) = II
3029, 6cnf 21031 . . . . . . . . . . . . . 14 (𝐴 ∈ (II Cn 𝑅) → 𝐴:(0[,]1)⟶ 𝑅)
3127, 30syl 17 . . . . . . . . . . . . 13 (𝜑𝐴:(0[,]1)⟶ 𝑅)
32 ffvelrn 6343 . . . . . . . . . . . . 13 ((𝐴:(0[,]1)⟶ 𝑅 ∧ 0 ∈ (0[,]1)) → (𝐴‘0) ∈ 𝑅)
3331, 17, 32sylancl 693 . . . . . . . . . . . 12 (𝜑 → (𝐴‘0) ∈ 𝑅)
344, 8, 33cnmptc 21446 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐴‘0)) ∈ (II Cn 𝑅))
3528, 34syl5eqel 2703 . . . . . . . . . 10 (𝜑 → ((0[,]1) × {(𝐴‘0)}) ∈ (II Cn 𝑅))
3627, 35phtpycn 22763 . . . . . . . . 9 (𝜑 → (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)})) ⊆ ((II ×t II) Cn 𝑅))
37 txsconn.7 . . . . . . . . 9 (𝜑𝐺 ∈ (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)})))
3836, 37sseldd 3596 . . . . . . . 8 (𝜑𝐺 ∈ ((II ×t II) Cn 𝑅))
39 iitop 22664 . . . . . . . . . 10 II ∈ Top
4039, 39, 29, 29txunii 21377 . . . . . . . . 9 ((0[,]1) × (0[,]1)) = (II ×t II)
4140, 6cnf 21031 . . . . . . . 8 (𝐺 ∈ ((II ×t II) Cn 𝑅) → 𝐺:((0[,]1) × (0[,]1))⟶ 𝑅)
42 ffn 6032 . . . . . . . 8 (𝐺:((0[,]1) × (0[,]1))⟶ 𝑅𝐺 Fn ((0[,]1) × (0[,]1)))
4338, 41, 423syl 18 . . . . . . 7 (𝜑𝐺 Fn ((0[,]1) × (0[,]1)))
44 fnov 6753 . . . . . . 7 (𝐺 Fn ((0[,]1) × (0[,]1)) ↔ 𝐺 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐺𝑦)))
4543, 44sylib 208 . . . . . 6 (𝜑𝐺 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐺𝑦)))
4645, 38eqeltrrd 2700 . . . . 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 21394 . . . . . . . . . . . . 13 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (2nd ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
498, 12, 48syl2anc 692 . . . . . . . . . . . 12 (𝜑 → (2nd ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
50 cnco 21051 . . . . . . . . . . . 12 ((𝐹 ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (2nd ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆)) → ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑆))
511, 49, 50syl2anc 692 . . . . . . . . . . 11 (𝜑 → ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑆))
5247, 51syl5eqel 2703 . . . . . . . . . 10 (𝜑𝐵 ∈ (II Cn 𝑆))
53 fconstmpt 5153 . . . . . . . . . . 11 ((0[,]1) × {(𝐵‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐵‘0))
5429, 10cnf 21031 . . . . . . . . . . . . . 14 (𝐵 ∈ (II Cn 𝑆) → 𝐵:(0[,]1)⟶ 𝑆)
5552, 54syl 17 . . . . . . . . . . . . 13 (𝜑𝐵:(0[,]1)⟶ 𝑆)
56 ffvelrn 6343 . . . . . . . . . . . . 13 ((𝐵:(0[,]1)⟶ 𝑆 ∧ 0 ∈ (0[,]1)) → (𝐵‘0) ∈ 𝑆)
5755, 17, 56sylancl 693 . . . . . . . . . . . 12 (𝜑 → (𝐵‘0) ∈ 𝑆)
584, 12, 57cnmptc 21446 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐵‘0)) ∈ (II Cn 𝑆))
5953, 58syl5eqel 2703 . . . . . . . . . 10 (𝜑 → ((0[,]1) × {(𝐵‘0)}) ∈ (II Cn 𝑆))
6052, 59phtpycn 22763 . . . . . . . . 9 (𝜑 → (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)})) ⊆ ((II ×t II) Cn 𝑆))
61 txsconn.8 . . . . . . . . 9 (𝜑𝐻 ∈ (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)})))
6260, 61sseldd 3596 . . . . . . . 8 (𝜑𝐻 ∈ ((II ×t II) Cn 𝑆))
6340, 10cnf 21031 . . . . . . . 8 (𝐻 ∈ ((II ×t II) Cn 𝑆) → 𝐻:((0[,]1) × (0[,]1))⟶ 𝑆)
64 ffn 6032 . . . . . . . 8 (𝐻:((0[,]1) × (0[,]1))⟶ 𝑆𝐻 Fn ((0[,]1) × (0[,]1)))
6562, 63, 643syl 18 . . . . . . 7 (𝜑𝐻 Fn ((0[,]1) × (0[,]1)))
66 fnov 6753 . . . . . . 7 (𝐻 Fn ((0[,]1) × (0[,]1)) ↔ 𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)))
6765, 66sylib 208 . . . . . 6 (𝜑𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)))
6867, 62eqeltrrd 2700 . . . . 5 (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)) ∈ ((II ×t II) Cn 𝑆))
694, 4, 46, 68cnmpt2t 21457 . . . 4 (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) ∈ ((II ×t II) Cn (𝑅 ×t 𝑆)))
7027, 35phtpyhtpy 22762 . . . . . . . . . 10 (𝜑 → (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)})) ⊆ (𝐴(II Htpy 𝑅)((0[,]1) × {(𝐴‘0)})))
7170, 37sseldd 3596 . . . . . . . . 9 (𝜑𝐺 ∈ (𝐴(II Htpy 𝑅)((0[,]1) × {(𝐴‘0)})))
724, 27, 35, 71htpyi 22754 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → ((𝑠𝐺0) = (𝐴𝑠) ∧ (𝑠𝐺1) = (((0[,]1) × {(𝐴‘0)})‘𝑠)))
7372simpld 475 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐺0) = (𝐴𝑠))
7422fveq1i 6179 . . . . . . . 8 (𝐴𝑠) = (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠)
75 fvco3 6262 . . . . . . . . 9 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
7616, 75sylan 488 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
7774, 76syl5eq 2666 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐴𝑠) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
78 ffvelrn 6343 . . . . . . . . 9 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (𝐹𝑠) ∈ ( 𝑅 × 𝑆))
7916, 78sylan 488 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹𝑠) ∈ ( 𝑅 × 𝑆))
80 fvres 6194 . . . . . . . 8 ((𝐹𝑠) ∈ ( 𝑅 × 𝑆) → ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)) = (1st ‘(𝐹𝑠)))
8179, 80syl 17 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)) = (1st ‘(𝐹𝑠)))
8273, 77, 813eqtrd 2658 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐺0) = (1st ‘(𝐹𝑠)))
8352, 59phtpyhtpy 22762 . . . . . . . . . 10 (𝜑 → (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)})) ⊆ (𝐵(II Htpy 𝑆)((0[,]1) × {(𝐵‘0)})))
8483, 61sseldd 3596 . . . . . . . . 9 (𝜑𝐻 ∈ (𝐵(II Htpy 𝑆)((0[,]1) × {(𝐵‘0)})))
854, 52, 59, 84htpyi 22754 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → ((𝑠𝐻0) = (𝐵𝑠) ∧ (𝑠𝐻1) = (((0[,]1) × {(𝐵‘0)})‘𝑠)))
8685simpld 475 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐵𝑠))
8747fveq1i 6179 . . . . . . . 8 (𝐵𝑠) = (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠)
88 fvco3 6262 . . . . . . . . 9 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
8916, 88sylan 488 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
9087, 89syl5eq 2666 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐵𝑠) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
91 fvres 6194 . . . . . . . 8 ((𝐹𝑠) ∈ ( 𝑅 × 𝑆) → ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)) = (2nd ‘(𝐹𝑠)))
9279, 91syl 17 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)) = (2nd ‘(𝐹𝑠)))
9386, 90, 923eqtrd 2658 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (2nd ‘(𝐹𝑠)))
9482, 93opeq12d 4401 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → ⟨(𝑠𝐺0), (𝑠𝐻0)⟩ = ⟨(1st ‘(𝐹𝑠)), (2nd ‘(𝐹𝑠))⟩)
95 simpr 477 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → 𝑠 ∈ (0[,]1))
96 oveq12 6644 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 0) → (𝑥𝐺𝑦) = (𝑠𝐺0))
97 oveq12 6644 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 0) → (𝑥𝐻𝑦) = (𝑠𝐻0))
9896, 97opeq12d 4401 . . . . . . 7 ((𝑥 = 𝑠𝑦 = 0) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
99 eqid 2620 . . . . . . 7 (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)
100 opex 4923 . . . . . . 7 ⟨(𝑠𝐺0), (𝑠𝐻0)⟩ ∈ V
10198, 99, 100ovmpt2a 6776 . . . . . 6 ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
10295, 17, 101sylancl 693 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
103 1st2nd2 7190 . . . . . 6 ((𝐹𝑠) ∈ ( 𝑅 × 𝑆) → (𝐹𝑠) = ⟨(1st ‘(𝐹𝑠)), (2nd ‘(𝐹𝑠))⟩)
10479, 103syl 17 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹𝑠) = ⟨(1st ‘(𝐹𝑠)), (2nd ‘(𝐹𝑠))⟩)
10594, 102, 1043eqtr4d 2664 . . . 4 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = (𝐹𝑠))
10672simprd 479 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐺1) = (((0[,]1) × {(𝐴‘0)})‘𝑠))
107 fvex 6188 . . . . . . . . 9 (𝐴‘0) ∈ V
108107fvconst2 6454 . . . . . . . 8 (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝐴‘0)})‘𝑠) = (𝐴‘0))
109108adantl 482 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐴‘0)})‘𝑠) = (𝐴‘0))
11022fveq1i 6179 . . . . . . . . 9 (𝐴‘0) = (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0)
111 fvco3 6262 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)))
11216, 17, 111sylancl 693 . . . . . . . . . 10 (𝜑 → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)))
113 fvres 6194 . . . . . . . . . . 11 ((𝐹‘0) ∈ ( 𝑅 × 𝑆) → ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)) = (1st ‘(𝐹‘0)))
11419, 113syl 17 . . . . . . . . . 10 (𝜑 → ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)) = (1st ‘(𝐹‘0)))
115112, 114eqtrd 2654 . . . . . . . . 9 (𝜑 → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = (1st ‘(𝐹‘0)))
116110, 115syl5eq 2666 . . . . . . . 8 (𝜑 → (𝐴‘0) = (1st ‘(𝐹‘0)))
117116adantr 481 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐴‘0) = (1st ‘(𝐹‘0)))
118106, 109, 1173eqtrd 2658 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐺1) = (1st ‘(𝐹‘0)))
11985simprd 479 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (((0[,]1) × {(𝐵‘0)})‘𝑠))
120 fvex 6188 . . . . . . . . 9 (𝐵‘0) ∈ V
121120fvconst2 6454 . . . . . . . 8 (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝐵‘0)})‘𝑠) = (𝐵‘0))
122121adantl 482 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐵‘0)})‘𝑠) = (𝐵‘0))
12347fveq1i 6179 . . . . . . . . 9 (𝐵‘0) = (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0)
124 fvco3 6262 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)))
12516, 17, 124sylancl 693 . . . . . . . . . 10 (𝜑 → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)))
126 fvres 6194 . . . . . . . . . . 11 ((𝐹‘0) ∈ ( 𝑅 × 𝑆) → ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)) = (2nd ‘(𝐹‘0)))
12719, 126syl 17 . . . . . . . . . 10 (𝜑 → ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)) = (2nd ‘(𝐹‘0)))
128125, 127eqtrd 2654 . . . . . . . . 9 (𝜑 → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = (2nd ‘(𝐹‘0)))
129123, 128syl5eq 2666 . . . . . . . 8 (𝜑 → (𝐵‘0) = (2nd ‘(𝐹‘0)))
130129adantr 481 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐵‘0) = (2nd ‘(𝐹‘0)))
131119, 122, 1303eqtrd 2658 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (2nd ‘(𝐹‘0)))
132118, 131opeq12d 4401 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → ⟨(𝑠𝐺1), (𝑠𝐻1)⟩ = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
133 1elunit 12276 . . . . . 6 1 ∈ (0[,]1)
134 oveq12 6644 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 1) → (𝑥𝐺𝑦) = (𝑠𝐺1))
135 oveq12 6644 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 1) → (𝑥𝐻𝑦) = (𝑠𝐻1))
136134, 135opeq12d 4401 . . . . . . 7 ((𝑥 = 𝑠𝑦 = 1) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
137 opex 4923 . . . . . . 7 ⟨(𝑠𝐺1), (𝑠𝐻1)⟩ ∈ V
138136, 99, 137ovmpt2a 6776 . . . . . 6 ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)1) = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
13995, 133, 138sylancl 693 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)1) = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
140 fvex 6188 . . . . . . . 8 (𝐹‘0) ∈ V
141140fvconst2 6454 . . . . . . 7 (𝑠 ∈ (0[,]1) → (((0[,]1) × {(𝐹‘0)})‘𝑠) = (𝐹‘0))
142141adantl 482 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐹‘0)})‘𝑠) = (𝐹‘0))
14319adantr 481 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘0) ∈ ( 𝑅 × 𝑆))
144 1st2nd2 7190 . . . . . . 7 ((𝐹‘0) ∈ ( 𝑅 × 𝑆) → (𝐹‘0) = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
145143, 144syl 17 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘0) = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
146142, 145eqtrd 2654 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐹‘0)})‘𝑠) = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
147132, 139, 1463eqtr4d 2664 . . . 4 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)1) = (((0[,]1) × {(𝐹‘0)})‘𝑠))
14827, 35, 37phtpyi 22764 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → ((0𝐺𝑠) = (𝐴‘0) ∧ (1𝐺𝑠) = (𝐴‘1)))
149148simpld 475 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐺𝑠) = (𝐴‘0))
150149, 117eqtrd 2654 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐺𝑠) = (1st ‘(𝐹‘0)))
15152, 59, 61phtpyi 22764 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → ((0𝐻𝑠) = (𝐵‘0) ∧ (1𝐻𝑠) = (𝐵‘1)))
152151simpld 475 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐵‘0))
153152, 130eqtrd 2654 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (2nd ‘(𝐹‘0)))
154150, 153opeq12d 4401 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → ⟨(0𝐺𝑠), (0𝐻𝑠)⟩ = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
155 oveq12 6644 . . . . . . . 8 ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝑥𝐺𝑦) = (0𝐺𝑠))
156 oveq12 6644 . . . . . . . 8 ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝑥𝐻𝑦) = (0𝐻𝑠))
157155, 156opeq12d 4401 . . . . . . 7 ((𝑥 = 0 ∧ 𝑦 = 𝑠) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
158 opex 4923 . . . . . . 7 ⟨(0𝐺𝑠), (0𝐻𝑠)⟩ ∈ V
159157, 99, 158ovmpt2a 6776 . . . . . 6 ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
16017, 95, 159sylancr 694 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
161154, 160, 1453eqtr4d 2664 . . . 4 ((𝜑𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = (𝐹‘0))
162148simprd 479 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐺𝑠) = (𝐴‘1))
16322fveq1i 6179 . . . . . . . . . 10 (𝐴‘1) = (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1)
164 fvco3 6262 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
16516, 133, 164sylancl 693 . . . . . . . . . 10 (𝜑 → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
166163, 165syl5eq 2666 . . . . . . . . 9 (𝜑 → (𝐴‘1) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
167 ffvelrn 6343 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ ( 𝑅 × 𝑆))
16816, 133, 167sylancl 693 . . . . . . . . . 10 (𝜑 → (𝐹‘1) ∈ ( 𝑅 × 𝑆))
169 fvres 6194 . . . . . . . . . 10 ((𝐹‘1) ∈ ( 𝑅 × 𝑆) → ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)) = (1st ‘(𝐹‘1)))
170168, 169syl 17 . . . . . . . . 9 (𝜑 → ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)) = (1st ‘(𝐹‘1)))
171166, 170eqtrd 2654 . . . . . . . 8 (𝜑 → (𝐴‘1) = (1st ‘(𝐹‘1)))
172171adantr 481 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐴‘1) = (1st ‘(𝐹‘1)))
173162, 172eqtrd 2654 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐺𝑠) = (1st ‘(𝐹‘1)))
174151simprd 479 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐵‘1))
17547fveq1i 6179 . . . . . . . . . 10 (𝐵‘1) = (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1)
176 fvco3 6262 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
17716, 133, 176sylancl 693 . . . . . . . . . 10 (𝜑 → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
178175, 177syl5eq 2666 . . . . . . . . 9 (𝜑 → (𝐵‘1) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
179 fvres 6194 . . . . . . . . . 10 ((𝐹‘1) ∈ ( 𝑅 × 𝑆) → ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)) = (2nd ‘(𝐹‘1)))
180168, 179syl 17 . . . . . . . . 9 (𝜑 → ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)) = (2nd ‘(𝐹‘1)))
181178, 180eqtrd 2654 . . . . . . . 8 (𝜑 → (𝐵‘1) = (2nd ‘(𝐹‘1)))
182181adantr 481 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐵‘1) = (2nd ‘(𝐹‘1)))
183174, 182eqtrd 2654 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (2nd ‘(𝐹‘1)))
184173, 183opeq12d 4401 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → ⟨(1𝐺𝑠), (1𝐻𝑠)⟩ = ⟨(1st ‘(𝐹‘1)), (2nd ‘(𝐹‘1))⟩)
185 oveq12 6644 . . . . . . . 8 ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑥𝐺𝑦) = (1𝐺𝑠))
186 oveq12 6644 . . . . . . . 8 ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑥𝐻𝑦) = (1𝐻𝑠))
187185, 186opeq12d 4401 . . . . . . 7 ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
188 opex 4923 . . . . . . 7 ⟨(1𝐺𝑠), (1𝐻𝑠)⟩ ∈ V
189187, 99, 188ovmpt2a 6776 . . . . . 6 ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
190133, 95, 189sylancr 694 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (1(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
191168adantr 481 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘1) ∈ ( 𝑅 × 𝑆))
192 1st2nd2 7190 . . . . . 6 ((𝐹‘1) ∈ ( 𝑅 × 𝑆) → (𝐹‘1) = ⟨(1st ‘(𝐹‘1)), (2nd ‘(𝐹‘1))⟩)
193191, 192syl 17 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹‘1) = ⟨(1st ‘(𝐹‘1)), (2nd ‘(𝐹‘1))⟩)
194184, 190, 1933eqtr4d 2664 . . . 4 ((𝜑𝑠 ∈ (0[,]1)) → (1(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = (𝐹‘1))
1951, 21, 69, 105, 147, 161, 194isphtpy2d 22767 . . 3 (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) ∈ (𝐹(PHtpy‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})))
196 ne0i 3913 . . 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)})) ≠ ∅)
197195, 196syl 17 . 2 (𝜑 → (𝐹(PHtpy‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})) ≠ ∅)
198 isphtpc 22774 . 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)})) ≠ ∅))
1991, 21, 197, 198syl3anbrc 1244 1 (𝜑𝐹( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wne 2791  c0 3907  {csn 4168  cop 4174   cuni 4427   class class class wbr 4644  cmpt 4720   × cxp 5102  cres 5106  ccom 5108   Fn wfn 5871  wf 5872  cfv 5876  (class class class)co 6635  cmpt2 6637  1st c1st 7151  2nd c2nd 7152  0cc0 9921  1c1 9922  [,]cicc 12163  Topctop 20679  TopOnctopon 20696   Cn ccn 21009   ×t ctx 21344  IIcii 22659   Htpy chtpy 22747  PHtpycphtpy 22748  phcphtpc 22749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-sup 8333  df-inf 8334  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-n0 11278  df-z 11363  df-uz 11673  df-q 11774  df-rp 11818  df-xneg 11931  df-xadd 11932  df-xmul 11933  df-icc 12167  df-seq 12785  df-exp 12844  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-topgen 16085  df-psmet 19719  df-xmet 19720  df-met 19721  df-bl 19722  df-mopn 19723  df-top 20680  df-topon 20697  df-bases 20731  df-cn 21012  df-cnp 21013  df-tx 21346  df-ii 22661  df-htpy 22750  df-phtpy 22751  df-phtpc 22772
This theorem is referenced by:  txsconn  31197
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