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Theorem txsconnlem 33834
Description: Lemma for txsconn 33835. (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 5694 . . 3 ((0[,]1) × {(𝐹‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐹‘0))
3 iitopon 24242 . . . . 5 II ∈ (TopOn‘(0[,]1))
43a1i 11 . . . 4 (𝜑 → II ∈ (TopOn‘(0[,]1)))
5 txsconn.1 . . . . . 6 (𝜑𝑅 ∈ Top)
6 eqid 2736 . . . . . . 7 𝑅 = 𝑅
76toptopon 22266 . . . . . 6 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘ 𝑅))
85, 7sylib 217 . . . . 5 (𝜑𝑅 ∈ (TopOn‘ 𝑅))
9 txsconn.2 . . . . . 6 (𝜑𝑆 ∈ Top)
10 eqid 2736 . . . . . . 7 𝑆 = 𝑆
1110toptopon 22266 . . . . . 6 (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘ 𝑆))
129, 11sylib 217 . . . . 5 (𝜑𝑆 ∈ (TopOn‘ 𝑆))
13 txtopon 22942 . . . . 5 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (𝑅 ×t 𝑆) ∈ (TopOn‘( 𝑅 × 𝑆)))
148, 12, 13syl2anc 584 . . . 4 (𝜑 → (𝑅 ×t 𝑆) ∈ (TopOn‘( 𝑅 × 𝑆)))
15 cnf2 22600 . . . . . 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 1371 . . . . 5 (𝜑𝐹:(0[,]1)⟶( 𝑅 × 𝑆))
17 0elunit 13386 . . . . 5 0 ∈ (0[,]1)
18 ffvelcdm 7032 . . . . 5 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ ( 𝑅 × 𝑆))
1916, 17, 18sylancl 586 . . . 4 (𝜑 → (𝐹‘0) ∈ ( 𝑅 × 𝑆))
204, 14, 19cnmptc 23013 . . 3 (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐹‘0)) ∈ (II Cn (𝑅 ×t 𝑆)))
212, 20eqeltrid 2842 . 2 (𝜑 → ((0[,]1) × {(𝐹‘0)}) ∈ (II Cn (𝑅 ×t 𝑆)))
22 txsconn.5 . . . . . . . . . . 11 𝐴 = ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)
23 tx1cn 22960 . . . . . . . . . . . . 13 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (1st ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
248, 12, 23syl2anc 584 . . . . . . . . . . . 12 (𝜑 → (1st ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
25 cnco 22617 . . . . . . . . . . . 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 584 . . . . . . . . . . 11 (𝜑 → ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑅))
2722, 26eqeltrid 2842 . . . . . . . . . 10 (𝜑𝐴 ∈ (II Cn 𝑅))
28 fconstmpt 5694 . . . . . . . . . . 11 ((0[,]1) × {(𝐴‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐴‘0))
29 iiuni 24244 . . . . . . . . . . . . . . 15 (0[,]1) = II
3029, 6cnf 22597 . . . . . . . . . . . . . 14 (𝐴 ∈ (II Cn 𝑅) → 𝐴:(0[,]1)⟶ 𝑅)
3127, 30syl 17 . . . . . . . . . . . . 13 (𝜑𝐴:(0[,]1)⟶ 𝑅)
32 ffvelcdm 7032 . . . . . . . . . . . . 13 ((𝐴:(0[,]1)⟶ 𝑅 ∧ 0 ∈ (0[,]1)) → (𝐴‘0) ∈ 𝑅)
3331, 17, 32sylancl 586 . . . . . . . . . . . 12 (𝜑 → (𝐴‘0) ∈ 𝑅)
344, 8, 33cnmptc 23013 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐴‘0)) ∈ (II Cn 𝑅))
3528, 34eqeltrid 2842 . . . . . . . . . 10 (𝜑 → ((0[,]1) × {(𝐴‘0)}) ∈ (II Cn 𝑅))
3627, 35phtpycn 24346 . . . . . . . . 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 3945 . . . . . . . 8 (𝜑𝐺 ∈ ((II ×t II) Cn 𝑅))
39 iitop 24243 . . . . . . . . . 10 II ∈ Top
4039, 39, 29, 29txunii 22944 . . . . . . . . 9 ((0[,]1) × (0[,]1)) = (II ×t II)
4140, 6cnf 22597 . . . . . . . 8 (𝐺 ∈ ((II ×t II) Cn 𝑅) → 𝐺:((0[,]1) × (0[,]1))⟶ 𝑅)
42 ffn 6668 . . . . . . . 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 7487 . . . . . . 7 (𝐺 Fn ((0[,]1) × (0[,]1)) ↔ 𝐺 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐺𝑦)))
4543, 44sylib 217 . . . . . 6 (𝜑𝐺 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐺𝑦)))
4645, 38eqeltrrd 2839 . . . . 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 22961 . . . . . . . . . . . . 13 ((𝑅 ∈ (TopOn‘ 𝑅) ∧ 𝑆 ∈ (TopOn‘ 𝑆)) → (2nd ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
498, 12, 48syl2anc 584 . . . . . . . . . . . 12 (𝜑 → (2nd ↾ ( 𝑅 × 𝑆)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
50 cnco 22617 . . . . . . . . . . . 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 584 . . . . . . . . . . 11 (𝜑 → ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹) ∈ (II Cn 𝑆))
5247, 51eqeltrid 2842 . . . . . . . . . 10 (𝜑𝐵 ∈ (II Cn 𝑆))
53 fconstmpt 5694 . . . . . . . . . . 11 ((0[,]1) × {(𝐵‘0)}) = (𝑥 ∈ (0[,]1) ↦ (𝐵‘0))
5429, 10cnf 22597 . . . . . . . . . . . . . 14 (𝐵 ∈ (II Cn 𝑆) → 𝐵:(0[,]1)⟶ 𝑆)
5552, 54syl 17 . . . . . . . . . . . . 13 (𝜑𝐵:(0[,]1)⟶ 𝑆)
56 ffvelcdm 7032 . . . . . . . . . . . . 13 ((𝐵:(0[,]1)⟶ 𝑆 ∧ 0 ∈ (0[,]1)) → (𝐵‘0) ∈ 𝑆)
5755, 17, 56sylancl 586 . . . . . . . . . . . 12 (𝜑 → (𝐵‘0) ∈ 𝑆)
584, 12, 57cnmptc 23013 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (0[,]1) ↦ (𝐵‘0)) ∈ (II Cn 𝑆))
5953, 58eqeltrid 2842 . . . . . . . . . 10 (𝜑 → ((0[,]1) × {(𝐵‘0)}) ∈ (II Cn 𝑆))
6052, 59phtpycn 24346 . . . . . . . . 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 3945 . . . . . . . 8 (𝜑𝐻 ∈ ((II ×t II) Cn 𝑆))
6340, 10cnf 22597 . . . . . . . 8 (𝐻 ∈ ((II ×t II) Cn 𝑆) → 𝐻:((0[,]1) × (0[,]1))⟶ 𝑆)
64 ffn 6668 . . . . . . . 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 7487 . . . . . . 7 (𝐻 Fn ((0[,]1) × (0[,]1)) ↔ 𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)))
6765, 66sylib 217 . . . . . 6 (𝜑𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)))
6867, 62eqeltrrd 2839 . . . . 5 (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻𝑦)) ∈ ((II ×t II) Cn 𝑆))
694, 4, 46, 68cnmpt2t 23024 . . . 4 (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) ∈ ((II ×t II) Cn (𝑅 ×t 𝑆)))
7027, 35phtpyhtpy 24345 . . . . . . . . . 10 (𝜑 → (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)})) ⊆ (𝐴(II Htpy 𝑅)((0[,]1) × {(𝐴‘0)})))
7170, 37sseldd 3945 . . . . . . . . 9 (𝜑𝐺 ∈ (𝐴(II Htpy 𝑅)((0[,]1) × {(𝐴‘0)})))
724, 27, 35, 71htpyi 24337 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → ((𝑠𝐺0) = (𝐴𝑠) ∧ (𝑠𝐺1) = (((0[,]1) × {(𝐴‘0)})‘𝑠)))
7372simpld 495 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐺0) = (𝐴𝑠))
7422fveq1i 6843 . . . . . . . 8 (𝐴𝑠) = (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠)
75 fvco3 6940 . . . . . . . . 9 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
7616, 75sylan 580 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
7774, 76eqtrid 2788 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐴𝑠) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
78 ffvelcdm 7032 . . . . . . . . 9 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (𝐹𝑠) ∈ ( 𝑅 × 𝑆))
7916, 78sylan 580 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹𝑠) ∈ ( 𝑅 × 𝑆))
80 fvres 6861 . . . . . . . 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 2780 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐺0) = (1st ‘(𝐹𝑠)))
8352, 59phtpyhtpy 24345 . . . . . . . . . 10 (𝜑 → (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)})) ⊆ (𝐵(II Htpy 𝑆)((0[,]1) × {(𝐵‘0)})))
8483, 61sseldd 3945 . . . . . . . . 9 (𝜑𝐻 ∈ (𝐵(II Htpy 𝑆)((0[,]1) × {(𝐵‘0)})))
854, 52, 59, 84htpyi 24337 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → ((𝑠𝐻0) = (𝐵𝑠) ∧ (𝑠𝐻1) = (((0[,]1) × {(𝐵‘0)})‘𝑠)))
8685simpld 495 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐵𝑠))
8747fveq1i 6843 . . . . . . . 8 (𝐵𝑠) = (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠)
88 fvco3 6940 . . . . . . . . 9 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 𝑠 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
8916, 88sylan 580 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘𝑠) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
9087, 89eqtrid 2788 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐵𝑠) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹𝑠)))
91 fvres 6861 . . . . . . . 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 2780 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (2nd ‘(𝐹𝑠)))
9482, 93opeq12d 4838 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → ⟨(𝑠𝐺0), (𝑠𝐻0)⟩ = ⟨(1st ‘(𝐹𝑠)), (2nd ‘(𝐹𝑠))⟩)
95 simpr 485 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → 𝑠 ∈ (0[,]1))
96 oveq12 7366 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 0) → (𝑥𝐺𝑦) = (𝑠𝐺0))
97 oveq12 7366 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 0) → (𝑥𝐻𝑦) = (𝑠𝐻0))
9896, 97opeq12d 4838 . . . . . . 7 ((𝑥 = 𝑠𝑦 = 0) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
99 eqid 2736 . . . . . . 7 (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)
100 opex 5421 . . . . . . 7 ⟨(𝑠𝐺0), (𝑠𝐻0)⟩ ∈ V
10198, 99, 100ovmpoa 7510 . . . . . 6 ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
10295, 17, 101sylancl 586 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = ⟨(𝑠𝐺0), (𝑠𝐻0)⟩)
103 1st2nd2 7960 . . . . . 6 ((𝐹𝑠) ∈ ( 𝑅 × 𝑆) → (𝐹𝑠) = ⟨(1st ‘(𝐹𝑠)), (2nd ‘(𝐹𝑠))⟩)
10479, 103syl 17 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (𝐹𝑠) = ⟨(1st ‘(𝐹𝑠)), (2nd ‘(𝐹𝑠))⟩)
10594, 102, 1043eqtr4d 2786 . . . 4 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)0) = (𝐹𝑠))
10672simprd 496 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐺1) = (((0[,]1) × {(𝐴‘0)})‘𝑠))
107 fvex 6855 . . . . . . . . 9 (𝐴‘0) ∈ V
108107fvconst2 7153 . . . . . . . 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 6843 . . . . . . . . 9 (𝐴‘0) = (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0)
111 fvco3 6940 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)))
11216, 17, 111sylancl 586 . . . . . . . . . 10 (𝜑 → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)))
113 fvres 6861 . . . . . . . . . . 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 2776 . . . . . . . . 9 (𝜑 → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = (1st ‘(𝐹‘0)))
116110, 115eqtrid 2788 . . . . . . . 8 (𝜑 → (𝐴‘0) = (1st ‘(𝐹‘0)))
117116adantr 481 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐴‘0) = (1st ‘(𝐹‘0)))
118106, 109, 1173eqtrd 2780 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐺1) = (1st ‘(𝐹‘0)))
11985simprd 496 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (((0[,]1) × {(𝐵‘0)})‘𝑠))
120 fvex 6855 . . . . . . . . 9 (𝐵‘0) ∈ V
121120fvconst2 7153 . . . . . . . 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 6843 . . . . . . . . 9 (𝐵‘0) = (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0)
124 fvco3 6940 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 0 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)))
12516, 17, 124sylancl 586 . . . . . . . . . 10 (𝜑 → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘0)))
126 fvres 6861 . . . . . . . . . . 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 2776 . . . . . . . . 9 (𝜑 → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘0) = (2nd ‘(𝐹‘0)))
129123, 128eqtrid 2788 . . . . . . . 8 (𝜑 → (𝐵‘0) = (2nd ‘(𝐹‘0)))
130129adantr 481 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐵‘0) = (2nd ‘(𝐹‘0)))
131119, 122, 1303eqtrd 2780 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (2nd ‘(𝐹‘0)))
132118, 131opeq12d 4838 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → ⟨(𝑠𝐺1), (𝑠𝐻1)⟩ = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
133 1elunit 13387 . . . . . 6 1 ∈ (0[,]1)
134 oveq12 7366 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 1) → (𝑥𝐺𝑦) = (𝑠𝐺1))
135 oveq12 7366 . . . . . . . 8 ((𝑥 = 𝑠𝑦 = 1) → (𝑥𝐻𝑦) = (𝑠𝐻1))
136134, 135opeq12d 4838 . . . . . . 7 ((𝑥 = 𝑠𝑦 = 1) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
137 opex 5421 . . . . . . 7 ⟨(𝑠𝐺1), (𝑠𝐻1)⟩ ∈ V
138136, 99, 137ovmpoa 7510 . . . . . 6 ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)1) = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
13995, 133, 138sylancl 586 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)1) = ⟨(𝑠𝐺1), (𝑠𝐻1)⟩)
140 fvex 6855 . . . . . . . 8 (𝐹‘0) ∈ V
141140fvconst2 7153 . . . . . . 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 7960 . . . . . . 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 2776 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (((0[,]1) × {(𝐹‘0)})‘𝑠) = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
147132, 139, 1463eqtr4d 2786 . . . 4 ((𝜑𝑠 ∈ (0[,]1)) → (𝑠(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)1) = (((0[,]1) × {(𝐹‘0)})‘𝑠))
14827, 35, 37phtpyi 24347 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → ((0𝐺𝑠) = (𝐴‘0) ∧ (1𝐺𝑠) = (𝐴‘1)))
149148simpld 495 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐺𝑠) = (𝐴‘0))
150149, 117eqtrd 2776 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐺𝑠) = (1st ‘(𝐹‘0)))
15152, 59, 61phtpyi 24347 . . . . . . . 8 ((𝜑𝑠 ∈ (0[,]1)) → ((0𝐻𝑠) = (𝐵‘0) ∧ (1𝐻𝑠) = (𝐵‘1)))
152151simpld 495 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐵‘0))
153152, 130eqtrd 2776 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (2nd ‘(𝐹‘0)))
154150, 153opeq12d 4838 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → ⟨(0𝐺𝑠), (0𝐻𝑠)⟩ = ⟨(1st ‘(𝐹‘0)), (2nd ‘(𝐹‘0))⟩)
155 oveq12 7366 . . . . . . . 8 ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝑥𝐺𝑦) = (0𝐺𝑠))
156 oveq12 7366 . . . . . . . 8 ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝑥𝐻𝑦) = (0𝐻𝑠))
157155, 156opeq12d 4838 . . . . . . 7 ((𝑥 = 0 ∧ 𝑦 = 𝑠) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
158 opex 5421 . . . . . . 7 ⟨(0𝐺𝑠), (0𝐻𝑠)⟩ ∈ V
159157, 99, 158ovmpoa 7510 . . . . . 6 ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
16017, 95, 159sylancr 587 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(0𝐺𝑠), (0𝐻𝑠)⟩)
161154, 160, 1453eqtr4d 2786 . . . 4 ((𝜑𝑠 ∈ (0[,]1)) → (0(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = (𝐹‘0))
162148simprd 496 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐺𝑠) = (𝐴‘1))
16322fveq1i 6843 . . . . . . . . . 10 (𝐴‘1) = (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1)
164 fvco3 6940 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
16516, 133, 164sylancl 586 . . . . . . . . . 10 (𝜑 → (((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
166163, 165eqtrid 2788 . . . . . . . . 9 (𝜑 → (𝐴‘1) = ((1st ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
167 ffvelcdm 7032 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ ( 𝑅 × 𝑆))
16816, 133, 167sylancl 586 . . . . . . . . . 10 (𝜑 → (𝐹‘1) ∈ ( 𝑅 × 𝑆))
169 fvres 6861 . . . . . . . . . 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 2776 . . . . . . . 8 (𝜑 → (𝐴‘1) = (1st ‘(𝐹‘1)))
172171adantr 481 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐴‘1) = (1st ‘(𝐹‘1)))
173162, 172eqtrd 2776 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐺𝑠) = (1st ‘(𝐹‘1)))
174151simprd 496 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐵‘1))
17547fveq1i 6843 . . . . . . . . . 10 (𝐵‘1) = (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1)
176 fvco3 6940 . . . . . . . . . . 11 ((𝐹:(0[,]1)⟶( 𝑅 × 𝑆) ∧ 1 ∈ (0[,]1)) → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
17716, 133, 176sylancl 586 . . . . . . . . . 10 (𝜑 → (((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)‘1) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
178175, 177eqtrid 2788 . . . . . . . . 9 (𝜑 → (𝐵‘1) = ((2nd ↾ ( 𝑅 × 𝑆))‘(𝐹‘1)))
179 fvres 6861 . . . . . . . . . 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 2776 . . . . . . . 8 (𝜑 → (𝐵‘1) = (2nd ‘(𝐹‘1)))
182181adantr 481 . . . . . . 7 ((𝜑𝑠 ∈ (0[,]1)) → (𝐵‘1) = (2nd ‘(𝐹‘1)))
183174, 182eqtrd 2776 . . . . . 6 ((𝜑𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (2nd ‘(𝐹‘1)))
184173, 183opeq12d 4838 . . . . 5 ((𝜑𝑠 ∈ (0[,]1)) → ⟨(1𝐺𝑠), (1𝐻𝑠)⟩ = ⟨(1st ‘(𝐹‘1)), (2nd ‘(𝐹‘1))⟩)
185 oveq12 7366 . . . . . . . 8 ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑥𝐺𝑦) = (1𝐺𝑠))
186 oveq12 7366 . . . . . . . 8 ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑥𝐻𝑦) = (1𝐻𝑠))
187185, 186opeq12d 4838 . . . . . . 7 ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩ = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
188 opex 5421 . . . . . . 7 ⟨(1𝐺𝑠), (1𝐻𝑠)⟩ ∈ V
189187, 99, 188ovmpoa 7510 . . . . . 6 ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = ⟨(1𝐺𝑠), (1𝐻𝑠)⟩)
190133, 95, 189sylancr 587 . . . . 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 7960 . . . . . 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 2786 . . . 4 ((𝜑𝑠 ∈ (0[,]1)) → (1(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩)𝑠) = (𝐹‘1))
1951, 21, 69, 105, 147, 161, 194isphtpy2d 24350 . . 3 (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ⟨(𝑥𝐺𝑦), (𝑥𝐻𝑦)⟩) ∈ (𝐹(PHtpy‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})))
196195ne0d 4295 . 2 (𝜑 → (𝐹(PHtpy‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})) ≠ ∅)
197 isphtpc 24357 . 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)})) ≠ ∅))
1981, 21, 196, 197syl3anbrc 1343 1 (𝜑𝐹( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2943  c0 4282  {csn 4586  cop 4592   cuni 4865   class class class wbr 5105  cmpt 5188   × cxp 5631  cres 5635  ccom 5637   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  0cc0 11051  1c1 11052  [,]cicc 13267  Topctop 22242  TopOnctopon 22259   Cn ccn 22575   ×t ctx 22911  IIcii 24238   Htpy chtpy 24330  PHtpycphtpy 24331  phcphtpc 24332
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-icc 13271  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cn 22578  df-cnp 22579  df-tx 22913  df-ii 24240  df-htpy 24333  df-phtpy 24334  df-phtpc 24355
This theorem is referenced by:  txsconn  33835
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