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Theorem cnmpt1t 23587
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 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
cnmpt11.a (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
cnmpt1t.b (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐿))
Assertion
Ref Expression
cnmpt1t (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ ⟨𝐴, 𝐡⟩) ∈ (𝐽 Cn (𝐾 Γ—t 𝐿)))
Distinct variable groups:   πœ‘,π‘₯   π‘₯,𝐽   π‘₯,𝑋   π‘₯,𝐾   π‘₯,𝐿
Allowed substitution hints:   𝐴(π‘₯)   𝐡(π‘₯)

Proof of Theorem cnmpt1t
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cnmptid.j . . . 4 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2 toponuni 22834 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
3 mpteq1 5243 . . . 4 (𝑋 = βˆͺ 𝐽 β†’ (π‘₯ ∈ 𝑋 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩) = (π‘₯ ∈ βˆͺ 𝐽 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩))
41, 2, 33syl 18 . . 3 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩) = (π‘₯ ∈ βˆͺ 𝐽 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩))
5 simpr 483 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
6 cnmpt11.a . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
7 cntop2 23163 . . . . . . . . . 10 ((π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) β†’ 𝐾 ∈ Top)
86, 7syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐾 ∈ Top)
9 toptopon2 22838 . . . . . . . . 9 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
108, 9sylib 217 . . . . . . . 8 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
11 cnf2 23171 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) ∧ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴):π‘‹βŸΆβˆͺ 𝐾)
121, 10, 6, 11syl3anc 1368 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴):π‘‹βŸΆβˆͺ 𝐾)
1312fvmptelcdm 7126 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ βˆͺ 𝐾)
14 eqid 2727 . . . . . . 7 (π‘₯ ∈ 𝑋 ↦ 𝐴) = (π‘₯ ∈ 𝑋 ↦ 𝐴)
1514fvmpt2 7019 . . . . . 6 ((π‘₯ ∈ 𝑋 ∧ 𝐴 ∈ βˆͺ 𝐾) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯) = 𝐴)
165, 13, 15syl2anc 582 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯) = 𝐴)
17 cnmpt1t.b . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐿))
18 cntop2 23163 . . . . . . . . . 10 ((π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐿) β†’ 𝐿 ∈ Top)
1917, 18syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐿 ∈ Top)
20 toptopon2 22838 . . . . . . . . 9 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
2119, 20sylib 217 . . . . . . . 8 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
22 cnf2 23171 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿) ∧ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐿)) β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡):π‘‹βŸΆβˆͺ 𝐿)
231, 21, 17, 22syl3anc 1368 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡):π‘‹βŸΆβˆͺ 𝐿)
2423fvmptelcdm 7126 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐡 ∈ βˆͺ 𝐿)
25 eqid 2727 . . . . . . 7 (π‘₯ ∈ 𝑋 ↦ 𝐡) = (π‘₯ ∈ 𝑋 ↦ 𝐡)
2625fvmpt2 7019 . . . . . 6 ((π‘₯ ∈ 𝑋 ∧ 𝐡 ∈ βˆͺ 𝐿) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯) = 𝐡)
275, 24, 26syl2anc 582 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯) = 𝐡)
2816, 27opeq12d 4884 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩ = ⟨𝐴, 𝐡⟩)
2928mpteq2dva 5250 . . 3 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩) = (π‘₯ ∈ 𝑋 ↦ ⟨𝐴, 𝐡⟩))
304, 29eqtr3d 2769 . 2 (πœ‘ β†’ (π‘₯ ∈ βˆͺ 𝐽 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩) = (π‘₯ ∈ 𝑋 ↦ ⟨𝐴, 𝐡⟩))
31 eqid 2727 . . . 4 βˆͺ 𝐽 = βˆͺ 𝐽
32 nfcv 2898 . . . . 5 β„²π‘¦βŸ¨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩
33 nffvmpt1 6911 . . . . . 6 β„²π‘₯((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘¦)
34 nffvmpt1 6911 . . . . . 6 β„²π‘₯((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘¦)
3533, 34nfop 4892 . . . . 5 β„²π‘₯⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘¦), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘¦)⟩
36 fveq2 6900 . . . . . 6 (π‘₯ = 𝑦 β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘¦))
37 fveq2 6900 . . . . . 6 (π‘₯ = 𝑦 β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘¦))
3836, 37opeq12d 4884 . . . . 5 (π‘₯ = 𝑦 β†’ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩ = ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘¦), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘¦)⟩)
3932, 35, 38cbvmpt 5261 . . . 4 (π‘₯ ∈ βˆͺ 𝐽 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩) = (𝑦 ∈ βˆͺ 𝐽 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘¦), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘¦)⟩)
4031, 39txcnmpt 23546 . . 3 (((π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) ∧ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐿)) β†’ (π‘₯ ∈ βˆͺ 𝐽 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩) ∈ (𝐽 Cn (𝐾 Γ—t 𝐿)))
416, 17, 40syl2anc 582 . 2 (πœ‘ β†’ (π‘₯ ∈ βˆͺ 𝐽 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩) ∈ (𝐽 Cn (𝐾 Γ—t 𝐿)))
4230, 41eqeltrrd 2829 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ ⟨𝐴, 𝐡⟩) ∈ (𝐽 Cn (𝐾 Γ—t 𝐿)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βŸ¨cop 4636  βˆͺ cuni 4910   ↦ cmpt 5233  βŸΆwf 6547  β€˜cfv 6551  (class class class)co 7424  Topctop 22813  TopOnctopon 22830   Cn ccn 23146   Γ—t ctx 23482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 7997  df-2nd 7998  df-map 8851  df-topgen 17430  df-top 22814  df-topon 22831  df-bases 22867  df-cn 23149  df-tx 23484
This theorem is referenced by:  cnmpt12f  23588  xkoinjcn  23609  txconn  23611  imasnopn  23612  imasncld  23613  imasncls  23614  ptunhmeo  23730  xkohmeo  23737  cnrehmeo  24896  cnrehmeoOLD  24897
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