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Theorem cnmpt21 21696
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 6817 . . . . . . . . . 10 (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = ((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑥, 𝑦⟩)
2 simprl 811 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → 𝑥𝑋)
3 simprr 813 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → 𝑦𝑌)
4 cnmpt21.j . . . . . . . . . . . . . . . 16 (𝜑𝐽 ∈ (TopOn‘𝑋))
5 cnmpt21.k . . . . . . . . . . . . . . . 16 (𝜑𝐾 ∈ (TopOn‘𝑌))
6 txtopon 21616 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
74, 5, 6syl2anc 696 . . . . . . . . . . . . . . 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 21275 . . . . . . . . . . . . . . 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 1477 . . . . . . . . . . . . . 14 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍)
12 eqid 2760 . . . . . . . . . . . . . . 15 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
1312fmpt2 7406 . . . . . . . . . . . . . 14 (∀𝑥𝑋𝑦𝑌 𝐴𝑍 ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍)
1411, 13sylibr 224 . . . . . . . . . . . . 13 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐴𝑍)
15 rsp2 3074 . . . . . . . . . . . . 13 (∀𝑥𝑋𝑦𝑌 𝐴𝑍 → ((𝑥𝑋𝑦𝑌) → 𝐴𝑍))
1614, 15syl 17 . . . . . . . . . . . 12 (𝜑 → ((𝑥𝑋𝑦𝑌) → 𝐴𝑍))
1716imp 444 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → 𝐴𝑍)
1812ovmpt4g 6949 . . . . . . . . . . 11 ((𝑥𝑋𝑦𝑌𝐴𝑍) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
192, 3, 17, 18syl3anc 1477 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
201, 19syl5eqr 2808 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑥, 𝑦⟩) = 𝐴)
2120fveq2d 6357 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ((𝑧𝑍𝐵)‘((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑥, 𝑦⟩)) = ((𝑧𝑍𝐵)‘𝐴))
22 cnmpt21.c . . . . . . . . . . 11 (𝑧 = 𝐴𝐵 = 𝐶)
2322eleq1d 2824 . . . . . . . . . 10 (𝑧 = 𝐴 → (𝐵 𝑀𝐶 𝑀))
24 cnmpt21.b . . . . . . . . . . . . . . 15 (𝜑 → (𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀))
25 cntop2 21267 . . . . . . . . . . . . . . 15 ((𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀) → 𝑀 ∈ Top)
2624, 25syl 17 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ Top)
27 eqid 2760 . . . . . . . . . . . . . . 15 𝑀 = 𝑀
2827toptopon 20944 . . . . . . . . . . . . . 14 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘ 𝑀))
2926, 28sylib 208 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ (TopOn‘ 𝑀))
30 cnf2 21275 . . . . . . . . . . . . 13 ((𝐿 ∈ (TopOn‘𝑍) ∧ 𝑀 ∈ (TopOn‘ 𝑀) ∧ (𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀)) → (𝑧𝑍𝐵):𝑍 𝑀)
318, 29, 24, 30syl3anc 1477 . . . . . . . . . . . 12 (𝜑 → (𝑧𝑍𝐵):𝑍 𝑀)
32 eqid 2760 . . . . . . . . . . . . 13 (𝑧𝑍𝐵) = (𝑧𝑍𝐵)
3332fmpt 6545 . . . . . . . . . . . 12 (∀𝑧𝑍 𝐵 𝑀 ↔ (𝑧𝑍𝐵):𝑍 𝑀)
3431, 33sylibr 224 . . . . . . . . . . 11 (𝜑 → ∀𝑧𝑍 𝐵 𝑀)
3534adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ∀𝑧𝑍 𝐵 𝑀)
3623, 35, 17rspcdva 3455 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → 𝐶 𝑀)
3722, 32fvmptg 6443 . . . . . . . . 9 ((𝐴𝑍𝐶 𝑀) → ((𝑧𝑍𝐵)‘𝐴) = 𝐶)
3817, 36, 37syl2anc 696 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ((𝑧𝑍𝐵)‘𝐴) = 𝐶)
3921, 38eqtrd 2794 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ((𝑧𝑍𝐵)‘((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑥, 𝑦⟩)) = 𝐶)
40 opelxpi 5305 . . . . . . . 8 ((𝑥𝑋𝑦𝑌) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
41 fvco3 6438 . . . . . . . 8 (((𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌)) → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑧𝑍𝐵)‘((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑥, 𝑦⟩)))
4211, 40, 41syl2an 495 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑧𝑍𝐵)‘((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑥, 𝑦⟩)))
43 df-ov 6817 . . . . . . . 8 (𝑥(𝑥𝑋, 𝑦𝑌𝐶)𝑦) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩)
44 eqid 2760 . . . . . . . . . 10 (𝑥𝑋, 𝑦𝑌𝐶) = (𝑥𝑋, 𝑦𝑌𝐶)
4544ovmpt4g 6949 . . . . . . . . 9 ((𝑥𝑋𝑦𝑌𝐶 𝑀) → (𝑥(𝑥𝑋, 𝑦𝑌𝐶)𝑦) = 𝐶)
462, 3, 36, 45syl3anc 1477 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → (𝑥(𝑥𝑋, 𝑦𝑌𝐶)𝑦) = 𝐶)
4743, 46syl5eqr 2808 . . . . . . 7 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩) = 𝐶)
4839, 42, 473eqtr4d 2804 . . . . . 6 ((𝜑 ∧ (𝑥𝑋𝑦𝑌)) → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩))
4948ralrimivva 3109 . . . . 5 (𝜑 → ∀𝑥𝑋𝑦𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩))
50 nfv 1992 . . . . . 6 𝑢𝑦𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩)
51 nfcv 2902 . . . . . . 7 𝑥𝑌
52 nfcv 2902 . . . . . . . . . 10 𝑥(𝑧𝑍𝐵)
53 nfmpt21 6888 . . . . . . . . . 10 𝑥(𝑥𝑋, 𝑦𝑌𝐴)
5452, 53nfco 5443 . . . . . . . . 9 𝑥((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))
55 nfcv 2902 . . . . . . . . 9 𝑥𝑢, 𝑣
5654, 55nffv 6360 . . . . . . . 8 𝑥(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩)
57 nfmpt21 6888 . . . . . . . . 9 𝑥(𝑥𝑋, 𝑦𝑌𝐶)
5857, 55nffv 6360 . . . . . . . 8 𝑥((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)
5956, 58nfeq 2914 . . . . . . 7 𝑥(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)
6051, 59nfral 3083 . . . . . 6 𝑥𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)
61 nfv 1992 . . . . . . . 8 𝑣(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩)
62 nfcv 2902 . . . . . . . . . . 11 𝑦(𝑧𝑍𝐵)
63 nfmpt22 6889 . . . . . . . . . . 11 𝑦(𝑥𝑋, 𝑦𝑌𝐴)
6462, 63nfco 5443 . . . . . . . . . 10 𝑦((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))
65 nfcv 2902 . . . . . . . . . 10 𝑦𝑥, 𝑣
6664, 65nffv 6360 . . . . . . . . 9 𝑦(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩)
67 nfmpt22 6889 . . . . . . . . . 10 𝑦(𝑥𝑋, 𝑦𝑌𝐶)
6867, 65nffv 6360 . . . . . . . . 9 𝑦((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩)
6966, 68nfeq 2914 . . . . . . . 8 𝑦(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩)
70 opeq2 4554 . . . . . . . . . 10 (𝑦 = 𝑣 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩)
7170fveq2d 6357 . . . . . . . . 9 (𝑦 = 𝑣 → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩))
7270fveq2d 6357 . . . . . . . . 9 (𝑦 = 𝑣 → ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩))
7371, 72eqeq12d 2775 . . . . . . . 8 (𝑦 = 𝑣 → ((((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩) ↔ (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩)))
7461, 69, 73cbvral 3306 . . . . . . 7 (∀𝑦𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩) ↔ ∀𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩))
75 opeq1 4553 . . . . . . . . . 10 (𝑥 = 𝑢 → ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩)
7675fveq2d 6357 . . . . . . . . 9 (𝑥 = 𝑢 → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩) = (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩))
7775fveq2d 6357 . . . . . . . . 9 (𝑥 = 𝑢 → ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩))
7876, 77eqeq12d 2775 . . . . . . . 8 (𝑥 = 𝑢 → ((((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩) ↔ (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)))
7978ralbidv 3124 . . . . . . 7 (𝑥 = 𝑢 → (∀𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑣⟩) ↔ ∀𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)))
8074, 79syl5bb 272 . . . . . 6 (𝑥 = 𝑢 → (∀𝑦𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩) ↔ ∀𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)))
8150, 60, 80cbvral 3306 . . . . 5 (∀𝑥𝑋𝑦𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑥, 𝑦⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑥, 𝑦⟩) ↔ ∀𝑢𝑋𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩))
8249, 81sylib 208 . . . 4 (𝜑 → ∀𝑢𝑋𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩))
83 fveq2 6353 . . . . . 6 (𝑤 = ⟨𝑢, 𝑣⟩ → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘𝑤) = (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩))
84 fveq2 6353 . . . . . 6 (𝑤 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑋, 𝑦𝑌𝐶)‘𝑤) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩))
8583, 84eqeq12d 2775 . . . . 5 (𝑤 = ⟨𝑢, 𝑣⟩ → ((((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘𝑤) = ((𝑥𝑋, 𝑦𝑌𝐶)‘𝑤) ↔ (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩)))
8685ralxp 5419 . . . 4 (∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘𝑤) = ((𝑥𝑋, 𝑦𝑌𝐶)‘𝑤) ↔ ∀𝑢𝑋𝑣𝑌 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘⟨𝑢, 𝑣⟩) = ((𝑥𝑋, 𝑦𝑌𝐶)‘⟨𝑢, 𝑣⟩))
8782, 86sylibr 224 . . 3 (𝜑 → ∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘𝑤) = ((𝑥𝑋, 𝑦𝑌𝐶)‘𝑤))
88 fco 6219 . . . . . 6 (((𝑧𝑍𝐵):𝑍 𝑀 ∧ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶𝑍) → ((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)):(𝑋 × 𝑌)⟶ 𝑀)
8931, 11, 88syl2anc 696 . . . . 5 (𝜑 → ((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)):(𝑋 × 𝑌)⟶ 𝑀)
90 ffn 6206 . . . . 5 (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)):(𝑋 × 𝑌)⟶ 𝑀 → ((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)) Fn (𝑋 × 𝑌))
9189, 90syl 17 . . . 4 (𝜑 → ((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)) Fn (𝑋 × 𝑌))
9236ralrimivva 3109 . . . . . 6 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐶 𝑀)
9344fmpt2 7406 . . . . . 6 (∀𝑥𝑋𝑦𝑌 𝐶 𝑀 ↔ (𝑥𝑋, 𝑦𝑌𝐶):(𝑋 × 𝑌)⟶ 𝑀)
9492, 93sylib 208 . . . . 5 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐶):(𝑋 × 𝑌)⟶ 𝑀)
95 ffn 6206 . . . . 5 ((𝑥𝑋, 𝑦𝑌𝐶):(𝑋 × 𝑌)⟶ 𝑀 → (𝑥𝑋, 𝑦𝑌𝐶) Fn (𝑋 × 𝑌))
9694, 95syl 17 . . . 4 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐶) Fn (𝑋 × 𝑌))
97 eqfnfv 6475 . . . 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)))
9891, 96, 97syl2anc 696 . . 3 (𝜑 → (((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)) = (𝑥𝑋, 𝑦𝑌𝐶) ↔ ∀𝑤 ∈ (𝑋 × 𝑌)(((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴))‘𝑤) = ((𝑥𝑋, 𝑦𝑌𝐶)‘𝑤)))
9987, 98mpbird 247 . 2 (𝜑 → ((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)) = (𝑥𝑋, 𝑦𝑌𝐶))
100 cnco 21292 . . 3 (((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ (𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀)) → ((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
1019, 24, 100syl2anc 696 . 2 (𝜑 → ((𝑧𝑍𝐵) ∘ (𝑥𝑋, 𝑦𝑌𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
10299, 101eqeltrrd 2840 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐶) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  cop 4327   cuni 4588  cmpt 4881   × cxp 5264  ccom 5270   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6814  cmpt2 6816  Topctop 20920  TopOnctopon 20937   Cn ccn 21250   ×t ctx 21585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027  df-topgen 16326  df-top 20921  df-topon 20938  df-bases 20972  df-cn 21253  df-tx 21587
This theorem is referenced by:  cnmpt21f  21697  xkofvcn  21709  xkohmeo  21840  qustgplem  22145  prdstmdd  22148  divcn  22892  htpycom  22996  htpycc  23000  reparphti  23017  pcocn  23037  pcohtpylem  23039  pcopt  23042  pcopt2  23043  pcoass  23044  pcorevlem  23046  dipcn  27905
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