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Theorem cnmpt21 23694
Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmpt21.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmpt21.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmpt21.a (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
cnmpt21.l (𝜑𝐿 ∈ (TopOn‘𝑍))
cnmpt21.b (𝜑 → (𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀))
cnmpt21.c (𝑧 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
cnmpt21 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐶) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐽   𝑥,𝑦,𝑧,𝐿   𝜑,𝑥,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑥,𝑀,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑧,𝐾   𝑥,𝑍,𝑦,𝑧   𝑥,𝐵,𝑦   𝑧,𝐶
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑧)   𝐶(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem cnmpt21
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 7433 . . . . . . . . . 10 (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = ((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑥, 𝑦⟩)
2 simprl 771 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → 𝑥𝑋)
3 simprr 773 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → 𝑦𝑌)
4 cnmpt21.j . . . . . . . . . . . . . . . 16 (𝜑𝐽 ∈ (TopOn‘𝑋))
5 cnmpt21.k . . . . . . . . . . . . . . . 16 (𝜑𝐾 ∈ (TopOn‘𝑌))
6 txtopon 23614 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
74, 5, 6syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
8 cnmpt21.l . . . . . . . . . . . . . . 15 (𝜑𝐿 ∈ (TopOn‘𝑍))
9 cnmpt21.a . . . . . . . . . . . . . . 15 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
10 cnf2 23272 . . . . . . . . . . . . . . 15 (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍)
117, 8, 9, 10syl3anc 1370 . . . . . . . . . . . . . 14 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍)
12 eqid 2734 . . . . . . . . . . . . . . 15 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
1312fmpo 8091 . . . . . . . . . . . . . 14 (∀𝑥𝑋𝑦𝑌 𝐴𝑍 ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍)
1411, 13sylibr 234 . . . . . . . . . . . . 13 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐴𝑍)
15 rsp2 3274 . . . . . . . . . . . . 13 (∀𝑥𝑋𝑦𝑌 𝐴𝑍 → ((𝑥𝑋𝑦𝑌) → 𝐴𝑍))
1614, 15syl 17 . . . . . . . . . . . 12 (𝜑 → ((𝑥𝑋𝑦𝑌) → 𝐴𝑍))
1716imp 406 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → 𝐴𝑍)
1812ovmpt4g 7579 . . . . . . . . . . 11 ((𝑥𝑋𝑦𝑌𝐴𝑍) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
192, 3, 17, 18syl3anc 1370 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
201, 19eqtr3id 2788 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑥, 𝑦⟩) = 𝐴)
2120fveq2d 6910 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ((𝑧𝑍𝐵)‘((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑥, 𝑦⟩)) = ((𝑧𝑍𝐵)‘𝐴))
22 eqid 2734 . . . . . . . . 9 (𝑧𝑍𝐵) = (𝑧𝑍𝐵)
23 cnmpt21.c . . . . . . . . 9 (𝑧 = 𝐴𝐵 = 𝐶)
2423eleq1d 2823 . . . . . . . . . 10 (𝑧 = 𝐴 → (𝐵 𝑀𝐶 𝑀))
25 cnmpt21.b . . . . . . . . . . . . . . 15 (𝜑 → (𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀))
26 cntop2 23264 . . . . . . . . . . . . . . 15 ((𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀) → 𝑀 ∈ Top)
2725, 26syl 17 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ Top)
28 toptopon2 22939 . . . . . . . . . . . . . 14 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘ 𝑀))
2927, 28sylib 218 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (TopOn‘ 𝑀))
30 cnf2 23272 . . . . . . . . . . . . 13 ((𝐿 ∈ (TopOn‘𝑍) ∧ 𝑀 ∈ (TopOn‘ 𝑀) ∧ (𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀)) → (𝑧𝑍𝐵):𝑍 𝑀)
318, 29, 25, 30syl3anc 1370 . . . . . . . . . . . 12 (𝜑 → (𝑧𝑍𝐵):𝑍 𝑀)
3222fmpt 7129 . . . . . . . . . . . 12 (∀𝑧𝑍 𝐵 𝑀 ↔ (𝑧𝑍𝐵):𝑍 𝑀)
3331, 32sylibr 234 . . . . . . . . . . 11 (𝜑 → ∀𝑧𝑍 𝐵 𝑀)
3433adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ∀𝑧𝑍 𝐵 𝑀)
3524, 34, 17rspcdva 3622 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → 𝐶 𝑀)
3622, 23, 17, 35fvmptd3 7038 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ((𝑧𝑍𝐵)‘𝐴) = 𝐶)
3721, 36eqtrd 2774 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ((𝑧𝑍𝐵)‘((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑥, 𝑦⟩)) = 𝐶)
38 opelxpi 5725 . . . . . . . 8 ((𝑥𝑋𝑦𝑌) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
39 fvco3 7007 . . . . . . . 8 (((𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌)) → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑧𝑍𝐵)‘((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑥, 𝑦⟩)))
4011, 38, 39syl2an 596 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑧𝑍𝐵)‘((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑥, 𝑦⟩)))
41 df-ov 7433 . . . . . . . 8 (𝑥(𝑥𝑋, 𝑦𝑌𝐶)𝑦) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩)
42 eqid 2734 . . . . . . . . . 10 (𝑥𝑋, 𝑦𝑌𝐶) = (𝑥𝑋, 𝑦𝑌𝐶)
4342ovmpt4g 7579 . . . . . . . . 9 ((𝑥𝑋𝑦𝑌𝐶 𝑀) → (𝑥(𝑥𝑋, 𝑦𝑌𝐶)𝑦) = 𝐶)
442, 3, 35, 43syl3anc 1370 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → (𝑥(𝑥𝑋, 𝑦𝑌𝐶)𝑦) = 𝐶)
4541, 44eqtr3id 2788 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩) = 𝐶)
4637, 40, 453eqtr4d 2784 . . . . . 6 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩))
4746ralrimivva 3199 . . . . 5 (𝜑 → ∀𝑥𝑋𝑦𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩))
48 nfv 1911 . . . . . 6 𝑢𝑦𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩)
49 nfcv 2902 . . . . . . 7 𝑥𝑌
50 nfcv 2902 . . . . . . . . . 10 𝑥(𝑧𝑍𝐵)
51 nfmpo1 7512 . . . . . . . . . 10 𝑥(𝑥𝑋, 𝑦𝑌𝐴)
5250, 51nfco 5878 . . . . . . . . 9 𝑥((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))
53 nfcv 2902 . . . . . . . . 9 𝑥𝑢, 𝑣
5452, 53nffv 6916 . . . . . . . 8 𝑥(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩)
55 nfmpo1 7512 . . . . . . . . 9 𝑥(𝑥𝑋, 𝑦𝑌𝐶)
5655, 53nffv 6916 . . . . . . . 8 𝑥((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)
5754, 56nfeq 2916 . . . . . . 7 𝑥(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)
5849, 57nfralw 3308 . . . . . 6 𝑥𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)
59 nfv 1911 . . . . . . . 8 𝑣(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩)
60 nfcv 2902 . . . . . . . . . . 11 𝑦(𝑧𝑍𝐵)
61 nfmpo2 7513 . . . . . . . . . . 11 𝑦(𝑥𝑋, 𝑦𝑌𝐴)
6260, 61nfco 5878 . . . . . . . . . 10 𝑦((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))
63 nfcv 2902 . . . . . . . . . 10 𝑦𝑥, 𝑣
6462, 63nffv 6916 . . . . . . . . 9 𝑦(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩)
65 nfmpo2 7513 . . . . . . . . . 10 𝑦(𝑥𝑋, 𝑦𝑌𝐶)
6665, 63nffv 6916 . . . . . . . . 9 𝑦((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩)
6764, 66nfeq 2916 . . . . . . . 8 𝑦(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩)
68 opeq2 4878 . . . . . . . . . 10 (𝑦 = 𝑣 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩)
6968fveq2d 6910 . . . . . . . . 9 (𝑦 = 𝑣 → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩))
7068fveq2d 6910 . . . . . . . . 9 (𝑦 = 𝑣 → ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩))
7169, 70eqeq12d 2750 . . . . . . . 8 (𝑦 = 𝑣 → ((((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩) ↔ (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩)))
7259, 67, 71cbvralw 3303 . . . . . . 7 (∀𝑦𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩) ↔ ∀𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩))
73 opeq1 4877 . . . . . . . . . 10 (𝑥 = 𝑢 → ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩)
7473fveq2d 6910 . . . . . . . . 9 (𝑥 = 𝑢 → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩) = (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩))
7573fveq2d 6910 . . . . . . . . 9 (𝑥 = 𝑢 → ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩))
7674, 75eqeq12d 2750 . . . . . . . 8 (𝑥 = 𝑢 → ((((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩) ↔ (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)))
7776ralbidv 3175 . . . . . . 7 (𝑥 = 𝑢 → (∀𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩) ↔ ∀𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)))
7872, 77bitrid 283 . . . . . 6 (𝑥 = 𝑢 → (∀𝑦𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩) ↔ ∀𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)))
7948, 58, 78cbvralw 3303 . . . . 5 (∀𝑥𝑋𝑦𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩) ↔ ∀𝑢𝑋𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩))
8047, 79sylib 218 . . . 4 (𝜑 → ∀𝑢𝑋𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩))
81 fveq2 6906 . . . . . 6 (𝑤 = ⟨𝑢, 𝑣⟩ → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘𝑤) = (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩))
82 fveq2 6906 . . . . . 6 (𝑤 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑋, 𝑦𝑌𝐶)‘𝑤) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩))
8381, 82eqeq12d 2750 . . . . 5 (𝑤 = ⟨𝑢, 𝑣⟩ → ((((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘𝑤) = ((𝑥𝑋, 𝑦𝑌𝐶)‘𝑤) ↔ (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)))
8483ralxp 5854 . . . 4 (∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘𝑤) = ((𝑥𝑋, 𝑦𝑌𝐶)‘𝑤) ↔ ∀𝑢𝑋𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩))
8580, 84sylibr 234 . . 3 (𝜑 → ∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘𝑤) = ((𝑥𝑋, 𝑦𝑌𝐶)‘𝑤))
86 fco 6760 . . . . . 6 (((𝑧𝑍𝐵):𝑍 𝑀 ∧ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍) → ((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)):(𝑋 × 𝑌)⟶ 𝑀)
8731, 11, 86syl2anc 584 . . . . 5 (𝜑 → ((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)):(𝑋 × 𝑌)⟶ 𝑀)
8887ffnd 6737 . . . 4 (𝜑 → ((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)) Fn (𝑋 × 𝑌))
8935ralrimivva 3199 . . . . . 6 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐶 𝑀)
9042fmpo 8091 . . . . . 6 (∀𝑥𝑋𝑦𝑌 𝐶 𝑀 ↔ (𝑥𝑋, 𝑦𝑌𝐶):(𝑋 × 𝑌)⟶ 𝑀)
9189, 90sylib 218 . . . . 5 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐶):(𝑋 × 𝑌)⟶ 𝑀)
9291ffnd 6737 . . . 4 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐶) Fn (𝑋 × 𝑌))
93 eqfnfv 7050 . . . 4 ((((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)) Fn (𝑋 × 𝑌) ∧ (𝑥𝑋, 𝑦𝑌𝐶) Fn (𝑋 × 𝑌)) → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)) = (𝑥𝑋, 𝑦𝑌𝐶) ↔ ∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘𝑤) = ((𝑥𝑋, 𝑦𝑌𝐶)‘𝑤)))
9488, 92, 93syl2anc 584 . . 3 (𝜑 → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)) = (𝑥𝑋, 𝑦𝑌𝐶) ↔ ∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘𝑤) = ((𝑥𝑋, 𝑦𝑌𝐶)‘𝑤)))
9585, 94mpbird 257 . 2 (𝜑 → ((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)) = (𝑥𝑋, 𝑦𝑌𝐶))
96 cnco 23289 . . 3 (((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ (𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀)) → ((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
979, 25, 96syl2anc 584 . 2 (𝜑 → ((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
9895, 97eqeltrrd 2839 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐶) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  cop 4636   cuni 4911  cmpt 5230   × cxp 5686  ccom 5692   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  cmpo 7432  Topctop 22914  TopOnctopon 22931   Cn ccn 23247   ×t ctx 23583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-map 8866  df-topgen 17489  df-top 22915  df-topon 22932  df-bases 22968  df-cn 23250  df-tx 23585
This theorem is referenced by:  cnmpt21f  23695  xkofvcn  23707  xkohmeo  23838  qustgplem  24144  prdstmdd  24147  divcnOLD  24903  divcn  24905  htpycom  25021  htpycc  25025  reparphti  25042  reparphtiOLD  25043  pcocn  25063  pcohtpylem  25065  pcopt  25068  pcopt2  25069  pcoass  25070  pcorevlem  25072  dipcn  30748
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