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Theorem cnmpt21 22274
 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 7143 . . . . . . . . . 10 (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = ((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑥, 𝑦⟩)
2 simprl 770 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → 𝑥𝑋)
3 simprr 772 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → 𝑦𝑌)
4 cnmpt21.j . . . . . . . . . . . . . . . 16 (𝜑𝐽 ∈ (TopOn‘𝑋))
5 cnmpt21.k . . . . . . . . . . . . . . . 16 (𝜑𝐾 ∈ (TopOn‘𝑌))
6 txtopon 22194 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
74, 5, 6syl2anc 587 . . . . . . . . . . . . . . 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 21852 . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . 14 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍)
12 eqid 2822 . . . . . . . . . . . . . . 15 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
1312fmpo 7752 . . . . . . . . . . . . . 14 (∀𝑥𝑋𝑦𝑌 𝐴𝑍 ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍)
1411, 13sylibr 237 . . . . . . . . . . . . 13 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐴𝑍)
15 rsp2 3202 . . . . . . . . . . . . 13 (∀𝑥𝑋𝑦𝑌 𝐴𝑍 → ((𝑥𝑋𝑦𝑌) → 𝐴𝑍))
1614, 15syl 17 . . . . . . . . . . . 12 (𝜑 → ((𝑥𝑋𝑦𝑌) → 𝐴𝑍))
1716imp 410 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → 𝐴𝑍)
1812ovmpt4g 7281 . . . . . . . . . . 11 ((𝑥𝑋𝑦𝑌𝐴𝑍) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
192, 3, 17, 18syl3anc 1368 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
201, 19syl5eqr 2871 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑥, 𝑦⟩) = 𝐴)
2120fveq2d 6656 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ((𝑧𝑍𝐵)‘((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑥, 𝑦⟩)) = ((𝑧𝑍𝐵)‘𝐴))
22 eqid 2822 . . . . . . . . 9 (𝑧𝑍𝐵) = (𝑧𝑍𝐵)
23 cnmpt21.c . . . . . . . . 9 (𝑧 = 𝐴𝐵 = 𝐶)
2423eleq1d 2898 . . . . . . . . . 10 (𝑧 = 𝐴 → (𝐵 𝑀𝐶 𝑀))
25 cnmpt21.b . . . . . . . . . . . . . . 15 (𝜑 → (𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀))
26 cntop2 21844 . . . . . . . . . . . . . . 15 ((𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀) → 𝑀 ∈ Top)
2725, 26syl 17 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ Top)
28 toptopon2 21521 . . . . . . . . . . . . . 14 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘ 𝑀))
2927, 28sylib 221 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (TopOn‘ 𝑀))
30 cnf2 21852 . . . . . . . . . . . . 13 ((𝐿 ∈ (TopOn‘𝑍) ∧ 𝑀 ∈ (TopOn‘ 𝑀) ∧ (𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀)) → (𝑧𝑍𝐵):𝑍 𝑀)
318, 29, 25, 30syl3anc 1368 . . . . . . . . . . . 12 (𝜑 → (𝑧𝑍𝐵):𝑍 𝑀)
3222fmpt 6856 . . . . . . . . . . . 12 (∀𝑧𝑍 𝐵 𝑀 ↔ (𝑧𝑍𝐵):𝑍 𝑀)
3331, 32sylibr 237 . . . . . . . . . . 11 (𝜑 → ∀𝑧𝑍 𝐵 𝑀)
3433adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ∀𝑧𝑍 𝐵 𝑀)
3524, 34, 17rspcdva 3600 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → 𝐶 𝑀)
3622, 23, 17, 35fvmptd3 6773 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ((𝑧𝑍𝐵)‘𝐴) = 𝐶)
3721, 36eqtrd 2857 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ((𝑧𝑍𝐵)‘((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑥, 𝑦⟩)) = 𝐶)
38 opelxpi 5569 . . . . . . . 8 ((𝑥𝑋𝑦𝑌) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
39 fvco3 6742 . . . . . . . 8 (((𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌)) → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑧𝑍𝐵)‘((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑥, 𝑦⟩)))
4011, 38, 39syl2an 598 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑧𝑍𝐵)‘((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑥, 𝑦⟩)))
41 df-ov 7143 . . . . . . . 8 (𝑥(𝑥𝑋, 𝑦𝑌𝐶)𝑦) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩)
42 eqid 2822 . . . . . . . . . 10 (𝑥𝑋, 𝑦𝑌𝐶) = (𝑥𝑋, 𝑦𝑌𝐶)
4342ovmpt4g 7281 . . . . . . . . 9 ((𝑥𝑋𝑦𝑌𝐶 𝑀) → (𝑥(𝑥𝑋, 𝑦𝑌𝐶)𝑦) = 𝐶)
442, 3, 35, 43syl3anc 1368 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → (𝑥(𝑥𝑋, 𝑦𝑌𝐶)𝑦) = 𝐶)
4541, 44syl5eqr 2871 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩) = 𝐶)
4637, 40, 453eqtr4d 2867 . . . . . 6 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩))
4746ralrimivva 3181 . . . . 5 (𝜑 → ∀𝑥𝑋𝑦𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩))
48 nfv 1915 . . . . . 6 𝑢𝑦𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩)
49 nfcv 2979 . . . . . . 7 𝑥𝑌
50 nfcv 2979 . . . . . . . . . 10 𝑥(𝑧𝑍𝐵)
51 nfmpo1 7218 . . . . . . . . . 10 𝑥(𝑥𝑋, 𝑦𝑌𝐴)
5250, 51nfco 5713 . . . . . . . . 9 𝑥((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))
53 nfcv 2979 . . . . . . . . 9 𝑥𝑢, 𝑣
5452, 53nffv 6662 . . . . . . . 8 𝑥(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩)
55 nfmpo1 7218 . . . . . . . . 9 𝑥(𝑥𝑋, 𝑦𝑌𝐶)
5655, 53nffv 6662 . . . . . . . 8 𝑥((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)
5754, 56nfeq 2992 . . . . . . 7 𝑥(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)
5849, 57nfralw 3214 . . . . . 6 𝑥𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)
59 nfv 1915 . . . . . . . 8 𝑣(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩)
60 nfcv 2979 . . . . . . . . . . 11 𝑦(𝑧𝑍𝐵)
61 nfmpo2 7219 . . . . . . . . . . 11 𝑦(𝑥𝑋, 𝑦𝑌𝐴)
6260, 61nfco 5713 . . . . . . . . . 10 𝑦((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))
63 nfcv 2979 . . . . . . . . . 10 𝑦𝑥, 𝑣
6462, 63nffv 6662 . . . . . . . . 9 𝑦(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩)
65 nfmpo2 7219 . . . . . . . . . 10 𝑦(𝑥𝑋, 𝑦𝑌𝐶)
6665, 63nffv 6662 . . . . . . . . 9 𝑦((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩)
6764, 66nfeq 2992 . . . . . . . 8 𝑦(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩)
68 opeq2 4778 . . . . . . . . . 10 (𝑦 = 𝑣 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩)
6968fveq2d 6656 . . . . . . . . 9 (𝑦 = 𝑣 → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩))
7068fveq2d 6656 . . . . . . . . 9 (𝑦 = 𝑣 → ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩))
7169, 70eqeq12d 2838 . . . . . . . 8 (𝑦 = 𝑣 → ((((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩) ↔ (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩)))
7259, 67, 71cbvralw 3415 . . . . . . 7 (∀𝑦𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩) ↔ ∀𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩))
73 opeq1 4776 . . . . . . . . . 10 (𝑥 = 𝑢 → ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩)
7473fveq2d 6656 . . . . . . . . 9 (𝑥 = 𝑢 → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩) = (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩))
7573fveq2d 6656 . . . . . . . . 9 (𝑥 = 𝑢 → ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩))
7674, 75eqeq12d 2838 . . . . . . . 8 (𝑥 = 𝑢 → ((((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩) ↔ (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)))
7776ralbidv 3187 . . . . . . 7 (𝑥 = 𝑢 → (∀𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩) ↔ ∀𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)))
7872, 77syl5bb 286 . . . . . 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 3415 . . . . 5 (∀𝑥𝑋𝑦𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩) ↔ ∀𝑢𝑋𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩))
8047, 79sylib 221 . . . 4 (𝜑 → ∀𝑢𝑋𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩))
81 fveq2 6652 . . . . . 6 (𝑤 = ⟨𝑢, 𝑣⟩ → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘𝑤) = (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩))
82 fveq2 6652 . . . . . 6 (𝑤 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑋, 𝑦𝑌𝐶)‘𝑤) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩))
8381, 82eqeq12d 2838 . . . . 5 (𝑤 = ⟨𝑢, 𝑣⟩ → ((((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘𝑤) = ((𝑥𝑋, 𝑦𝑌𝐶)‘𝑤) ↔ (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)))
8483ralxp 5689 . . . 4 (∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘𝑤) = ((𝑥𝑋, 𝑦𝑌𝐶)‘𝑤) ↔ ∀𝑢𝑋𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩))
8580, 84sylibr 237 . . 3 (𝜑 → ∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘𝑤) = ((𝑥𝑋, 𝑦𝑌𝐶)‘𝑤))
86 fco 6512 . . . . . 6 (((𝑧𝑍𝐵):𝑍 𝑀 ∧ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍) → ((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)):(𝑋 × 𝑌)⟶ 𝑀)
8731, 11, 86syl2anc 587 . . . . 5 (𝜑 → ((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)):(𝑋 × 𝑌)⟶ 𝑀)
8887ffnd 6495 . . . 4 (𝜑 → ((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)) Fn (𝑋 × 𝑌))
8935ralrimivva 3181 . . . . . 6 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐶 𝑀)
9042fmpo 7752 . . . . . 6 (∀𝑥𝑋𝑦𝑌 𝐶 𝑀 ↔ (𝑥𝑋, 𝑦𝑌𝐶):(𝑋 × 𝑌)⟶ 𝑀)
9189, 90sylib 221 . . . . 5 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐶):(𝑋 × 𝑌)⟶ 𝑀)
9291ffnd 6495 . . . 4 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐶) Fn (𝑋 × 𝑌))
93 eqfnfv 6784 . . . 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 587 . . 3 (𝜑 → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)) = (𝑥𝑋, 𝑦𝑌𝐶) ↔ ∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘𝑤) = ((𝑥𝑋, 𝑦𝑌𝐶)‘𝑤)))
9585, 94mpbird 260 . 2 (𝜑 → ((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)) = (𝑥𝑋, 𝑦𝑌𝐶))
96 cnco 21869 . . 3 (((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ (𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀)) → ((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
979, 25, 96syl2anc 587 . 2 (𝜑 → ((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
9895, 97eqeltrrd 2915 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐶) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2114  ∀wral 3130  ⟨cop 4545  ∪ cuni 4813   ↦ cmpt 5122   × cxp 5530   ∘ ccom 5536   Fn wfn 6329  ⟶wf 6330  ‘cfv 6334  (class class class)co 7140   ∈ cmpo 7142  Topctop 21496  TopOnctopon 21513   Cn ccn 21827   ×t ctx 22163 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-map 8395  df-topgen 16708  df-top 21497  df-topon 21514  df-bases 21549  df-cn 21830  df-tx 22165 This theorem is referenced by:  cnmpt21f  22275  xkofvcn  22287  xkohmeo  22418  qustgplem  22724  prdstmdd  22727  divcn  23471  htpycom  23579  htpycc  23583  reparphti  23600  pcocn  23620  pcohtpylem  23622  pcopt  23625  pcopt2  23626  pcoass  23627  pcorevlem  23629  dipcn  28501
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