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Theorem cnmpt1t 21670
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 20921 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
3 mpteq1 4889 . . . 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 479 . . . . . 6 ((𝜑𝑥𝑋) → 𝑥𝑋)
6 cnmpt11.a . . . . . . . . . . 11 (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))
7 cntop2 21247 . . . . . . . . . . 11 ((𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
86, 7syl 17 . . . . . . . . . 10 (𝜑𝐾 ∈ Top)
9 eqid 2760 . . . . . . . . . . 11 𝐾 = 𝐾
109toptopon 20924 . . . . . . . . . 10 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
118, 10sylib 208 . . . . . . . . 9 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
12 cnf2 21255 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾)) → (𝑥𝑋𝐴):𝑋 𝐾)
131, 11, 6, 12syl3anc 1477 . . . . . . . 8 (𝜑 → (𝑥𝑋𝐴):𝑋 𝐾)
14 eqid 2760 . . . . . . . . 9 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
1514fmpt 6544 . . . . . . . 8 (∀𝑥𝑋 𝐴 𝐾 ↔ (𝑥𝑋𝐴):𝑋 𝐾)
1613, 15sylibr 224 . . . . . . 7 (𝜑 → ∀𝑥𝑋 𝐴 𝐾)
1716r19.21bi 3070 . . . . . 6 ((𝜑𝑥𝑋) → 𝐴 𝐾)
1814fvmpt2 6453 . . . . . 6 ((𝑥𝑋𝐴 𝐾) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
195, 17, 18syl2anc 696 . . . . 5 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
20 cnmpt1t.b . . . . . . . . . . 11 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))
21 cntop2 21247 . . . . . . . . . . 11 ((𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
2220, 21syl 17 . . . . . . . . . 10 (𝜑𝐿 ∈ Top)
23 eqid 2760 . . . . . . . . . . 11 𝐿 = 𝐿
2423toptopon 20924 . . . . . . . . . 10 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
2522, 24sylib 208 . . . . . . . . 9 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
26 cnf2 21255 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥𝑋𝐵):𝑋 𝐿)
271, 25, 20, 26syl3anc 1477 . . . . . . . 8 (𝜑 → (𝑥𝑋𝐵):𝑋 𝐿)
28 eqid 2760 . . . . . . . . 9 (𝑥𝑋𝐵) = (𝑥𝑋𝐵)
2928fmpt 6544 . . . . . . . 8 (∀𝑥𝑋 𝐵 𝐿 ↔ (𝑥𝑋𝐵):𝑋 𝐿)
3027, 29sylibr 224 . . . . . . 7 (𝜑 → ∀𝑥𝑋 𝐵 𝐿)
3130r19.21bi 3070 . . . . . 6 ((𝜑𝑥𝑋) → 𝐵 𝐿)
3228fvmpt2 6453 . . . . . 6 ((𝑥𝑋𝐵 𝐿) → ((𝑥𝑋𝐵)‘𝑥) = 𝐵)
335, 31, 32syl2anc 696 . . . . 5 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐵)‘𝑥) = 𝐵)
3419, 33opeq12d 4561 . . . 4 ((𝜑𝑥𝑋) → ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
3534mpteq2dva 4896 . . 3 (𝜑 → (𝑥𝑋 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
364, 35eqtr3d 2796 . 2 (𝜑 → (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
37 eqid 2760 . . . 4 𝐽 = 𝐽
38 nfcv 2902 . . . . 5 𝑦⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩
39 nffvmpt1 6360 . . . . . 6 𝑥((𝑥𝑋𝐴)‘𝑦)
40 nffvmpt1 6360 . . . . . 6 𝑥((𝑥𝑋𝐵)‘𝑦)
4139, 40nfop 4569 . . . . 5 𝑥⟨((𝑥𝑋𝐴)‘𝑦), ((𝑥𝑋𝐵)‘𝑦)⟩
42 fveq2 6352 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝑋𝐴)‘𝑥) = ((𝑥𝑋𝐴)‘𝑦))
43 fveq2 6352 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝑋𝐵)‘𝑥) = ((𝑥𝑋𝐵)‘𝑦))
4442, 43opeq12d 4561 . . . . 5 (𝑥 = 𝑦 → ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ = ⟨((𝑥𝑋𝐴)‘𝑦), ((𝑥𝑋𝐵)‘𝑦)⟩)
4538, 41, 44cbvmpt 4901 . . . 4 (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) = (𝑦 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑦), ((𝑥𝑋𝐵)‘𝑦)⟩)
4637, 45txcnmpt 21629 . . 3 (((𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾) ∧ (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿)) → (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
476, 20, 46syl2anc 696 . 2 (𝜑 → (𝑥 𝐽 ↦ ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
4836, 47eqeltrrd 2840 1 (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  cop 4327   cuni 4588  cmpt 4881  wf 6045  cfv 6049  (class class class)co 6813  Topctop 20900  TopOnctopon 20917   Cn ccn 21230   ×t ctx 21565
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 7114
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 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025  df-topgen 16306  df-top 20901  df-topon 20918  df-bases 20952  df-cn 21233  df-tx 21567
This theorem is referenced by:  cnmpt12f  21671  xkoinjcn  21692  txconn  21694  imasnopn  21695  imasncld  21696  imasncls  21697  ptunhmeo  21813  xkohmeo  21820  cnrehmeo  22953
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