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Theorem cnmpt1t 23039
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 22286 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
3 mpteq1 5202 . . . 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 486 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
6 cnmpt11.a . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
7 cntop2 22615 . . . . . . . . . 10 ((π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) β†’ 𝐾 ∈ Top)
86, 7syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐾 ∈ Top)
9 toptopon2 22290 . . . . . . . . 9 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
108, 9sylib 217 . . . . . . . 8 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
11 cnf2 22623 . . . . . . . 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 1372 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴):π‘‹βŸΆβˆͺ 𝐾)
1312fvmptelcdm 7065 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ βˆͺ 𝐾)
14 eqid 2733 . . . . . . 7 (π‘₯ ∈ 𝑋 ↦ 𝐴) = (π‘₯ ∈ 𝑋 ↦ 𝐴)
1514fvmpt2 6963 . . . . . 6 ((π‘₯ ∈ 𝑋 ∧ 𝐴 ∈ βˆͺ 𝐾) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯) = 𝐴)
165, 13, 15syl2anc 585 . . . . 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 22615 . . . . . . . . . 10 ((π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐿) β†’ 𝐿 ∈ Top)
1917, 18syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐿 ∈ Top)
20 toptopon2 22290 . . . . . . . . 9 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
2119, 20sylib 217 . . . . . . . 8 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
22 cnf2 22623 . . . . . . . 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 1372 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡):π‘‹βŸΆβˆͺ 𝐿)
2423fvmptelcdm 7065 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐡 ∈ βˆͺ 𝐿)
25 eqid 2733 . . . . . . 7 (π‘₯ ∈ 𝑋 ↦ 𝐡) = (π‘₯ ∈ 𝑋 ↦ 𝐡)
2625fvmpt2 6963 . . . . . 6 ((π‘₯ ∈ 𝑋 ∧ 𝐡 ∈ βˆͺ 𝐿) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯) = 𝐡)
275, 24, 26syl2anc 585 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯) = 𝐡)
2816, 27opeq12d 4842 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩ = ⟨𝐴, 𝐡⟩)
2928mpteq2dva 5209 . . 3 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩) = (π‘₯ ∈ 𝑋 ↦ ⟨𝐴, 𝐡⟩))
304, 29eqtr3d 2775 . 2 (πœ‘ β†’ (π‘₯ ∈ βˆͺ 𝐽 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩) = (π‘₯ ∈ 𝑋 ↦ ⟨𝐴, 𝐡⟩))
31 eqid 2733 . . . 4 βˆͺ 𝐽 = βˆͺ 𝐽
32 nfcv 2904 . . . . 5 β„²π‘¦βŸ¨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩
33 nffvmpt1 6857 . . . . . 6 β„²π‘₯((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘¦)
34 nffvmpt1 6857 . . . . . 6 β„²π‘₯((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘¦)
3533, 34nfop 4850 . . . . 5 β„²π‘₯⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘¦), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘¦)⟩
36 fveq2 6846 . . . . . 6 (π‘₯ = 𝑦 β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘¦))
37 fveq2 6846 . . . . . 6 (π‘₯ = 𝑦 β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘¦))
3836, 37opeq12d 4842 . . . . 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 5220 . . . 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 22998 . . 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 585 . 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 2835 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 397   = wceq 1542   ∈ wcel 2107  βŸ¨cop 4596  βˆͺ cuni 4869   ↦ cmpt 5192  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  Topctop 22265  TopOnctopon 22282   Cn ccn 22598   Γ—t ctx 22934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-map 8773  df-topgen 17333  df-top 22266  df-topon 22283  df-bases 22319  df-cn 22601  df-tx 22936
This theorem is referenced by:  cnmpt12f  23040  xkoinjcn  23061  txconn  23063  imasnopn  23064  imasncld  23065  imasncls  23066  ptunhmeo  23182  xkohmeo  23189  cnrehmeo  24339
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