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Theorem cnmpt1t 21378
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
cnmptid.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmpt11.a (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))
cnmpt1t.b (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))
Assertion
Ref Expression
cnmpt1t (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
Distinct variable groups:   𝜑,𝑥   𝑥,𝐽   𝑥,𝑋   𝑥,𝐾   𝑥,𝐿
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem cnmpt1t
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cnmptid.j . . . 4 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 toponuni 20642 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
3 mpteq1 4697 . . . 4 (𝑋 = 𝐽 → (𝑥𝑋 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩))
41, 2, 33syl 18 . . 3 (𝜑 → (𝑥𝑋 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩))
5 simpr 477 . . . . . 6 ((𝜑𝑥𝑋) → 𝑥𝑋)
6 cnmpt11.a . . . . . . . . . . 11 (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))
7 cntop2 20955 . . . . . . . . . . 11 ((𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
86, 7syl 17 . . . . . . . . . 10 (𝜑𝐾 ∈ Top)
9 eqid 2621 . . . . . . . . . . 11 𝐾 = 𝐾
109toptopon 20648 . . . . . . . . . 10 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
118, 10sylib 208 . . . . . . . . 9 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
12 cnf2 20963 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥𝑋𝐴):𝑋 𝐾)
131, 11, 6, 12syl3anc 1323 . . . . . . . 8 (𝜑 → (𝑥𝑋𝐴):𝑋 𝐾)
14 eqid 2621 . . . . . . . . 9 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
1514fmpt 6337 . . . . . . . 8 (∀𝑥𝑋 𝐴 𝐾 ↔ (𝑥𝑋𝐴):𝑋 𝐾)
1613, 15sylibr 224 . . . . . . 7 (𝜑 → ∀𝑥𝑋 𝐴 𝐾)
1716r19.21bi 2927 . . . . . 6 ((𝜑𝑥𝑋) → 𝐴 𝐾)
1814fvmpt2 6248 . . . . . 6 ((𝑥𝑋𝐴 𝐾) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
195, 17, 18syl2anc 692 . . . . 5 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
20 cnmpt1t.b . . . . . . . . . . 11 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))
21 cntop2 20955 . . . . . . . . . . 11 ((𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
2220, 21syl 17 . . . . . . . . . 10 (𝜑𝐿 ∈ Top)
23 eqid 2621 . . . . . . . . . . 11 𝐿 = 𝐿
2423toptopon 20648 . . . . . . . . . 10 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
2522, 24sylib 208 . . . . . . . . 9 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
26 cnf2 20963 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥𝑋𝐵):𝑋 𝐿)
271, 25, 20, 26syl3anc 1323 . . . . . . . 8 (𝜑 → (𝑥𝑋𝐵):𝑋 𝐿)
28 eqid 2621 . . . . . . . . 9 (𝑥𝑋𝐵) = (𝑥𝑋𝐵)
2928fmpt 6337 . . . . . . . 8 (∀𝑥𝑋 𝐵 𝐿 ↔ (𝑥𝑋𝐵):𝑋 𝐿)
3027, 29sylibr 224 . . . . . . 7 (𝜑 → ∀𝑥𝑋 𝐵 𝐿)
3130r19.21bi 2927 . . . . . 6 ((𝜑𝑥𝑋) → 𝐵 𝐿)
3228fvmpt2 6248 . . . . . 6 ((𝑥𝑋𝐵 𝐿) → ((𝑥𝑋𝐵)‘𝑥) = 𝐵)
335, 31, 32syl2anc 692 . . . . 5 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐵)‘𝑥) = 𝐵)
3419, 33opeq12d 4378 . . . 4 ((𝜑𝑥𝑋) → ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
3534mpteq2dva 4704 . . 3 (𝜑 → (𝑥𝑋 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
364, 35eqtr3d 2657 . 2 (𝜑 → (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
37 eqid 2621 . . . 4 𝐽 = 𝐽
38 nfcv 2761 . . . . 5 𝑦⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩
39 nffvmpt1 6156 . . . . . 6 𝑥((𝑥𝑋𝐴)‘𝑦)
40 nffvmpt1 6156 . . . . . 6 𝑥((𝑥𝑋𝐵)‘𝑦)
4139, 40nfop 4386 . . . . 5 𝑥⟨((𝑥𝑋𝐴)‘𝑦), ((𝑥𝑋𝐵)‘𝑦)⟩
42 fveq2 6148 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝑋𝐴)‘𝑥) = ((𝑥𝑋𝐴)‘𝑦))
43 fveq2 6148 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝑋𝐵)‘𝑥) = ((𝑥𝑋𝐵)‘𝑦))
4442, 43opeq12d 4378 . . . . 5 (𝑥 = 𝑦 → ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ = ⟨((𝑥𝑋𝐴)‘𝑦), ((𝑥𝑋𝐵)‘𝑦)⟩)
4538, 41, 44cbvmpt 4709 . . . 4 (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑦 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑦), ((𝑥𝑋𝐵)‘𝑦)⟩)
4637, 45txcnmpt 21337 . . 3 (((𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾) ∧ (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
476, 20, 46syl2anc 692 . 2 (𝜑 → (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
4836, 47eqeltrrd 2699 1 (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  cop 4154   cuni 4402  cmpt 4673  wf 5843  cfv 5847  (class class class)co 6604  Topctop 20617  TopOnctopon 20618   Cn ccn 20938   ×t ctx 21273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-cn 20941  df-tx 21275
This theorem is referenced by:  cnmpt12f  21379  xkoinjcn  21400  txconn  21402  imasnopn  21403  imasncld  21404  imasncls  21405  ptunhmeo  21521  xkohmeo  21528  cnrehmeo  22660
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