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Theorem cnmpt11f 23588
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))
cnmpt11f.f (πœ‘ β†’ 𝐹 ∈ (𝐾 Cn 𝐿))
Assertion
Ref Expression
cnmpt11f (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (πΉβ€˜π΄)) ∈ (𝐽 Cn 𝐿))
Distinct variable groups:   π‘₯,𝐹   πœ‘,π‘₯   π‘₯,𝐽   π‘₯,𝑋   π‘₯,𝐾   π‘₯,𝐿
Allowed substitution hint:   𝐴(π‘₯)

Proof of Theorem cnmpt11f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cnmptid.j . 2 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2 cnmpt11.a . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
3 cntop2 23165 . . . 4 ((π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) β†’ 𝐾 ∈ Top)
42, 3syl 17 . . 3 (πœ‘ β†’ 𝐾 ∈ Top)
5 toptopon2 22840 . . 3 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
64, 5sylib 217 . 2 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
7 cnmpt11f.f . . . . 5 (πœ‘ β†’ 𝐹 ∈ (𝐾 Cn 𝐿))
8 eqid 2728 . . . . . 6 βˆͺ 𝐾 = βˆͺ 𝐾
9 eqid 2728 . . . . . 6 βˆͺ 𝐿 = βˆͺ 𝐿
108, 9cnf 23170 . . . . 5 (𝐹 ∈ (𝐾 Cn 𝐿) β†’ 𝐹:βˆͺ 𝐾⟢βˆͺ 𝐿)
117, 10syl 17 . . . 4 (πœ‘ β†’ 𝐹:βˆͺ 𝐾⟢βˆͺ 𝐿)
1211feqmptd 6972 . . 3 (πœ‘ β†’ 𝐹 = (𝑦 ∈ βˆͺ 𝐾 ↦ (πΉβ€˜π‘¦)))
1312, 7eqeltrrd 2830 . 2 (πœ‘ β†’ (𝑦 ∈ βˆͺ 𝐾 ↦ (πΉβ€˜π‘¦)) ∈ (𝐾 Cn 𝐿))
14 fveq2 6902 . 2 (𝑦 = 𝐴 β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π΄))
151, 2, 6, 13, 14cnmpt11 23587 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (πΉβ€˜π΄)) ∈ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2098  βˆͺ cuni 4912   ↦ cmpt 5235  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426  Topctop 22815  TopOnctopon 22832   Cn ccn 23148
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 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
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 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-map 8853  df-top 22816  df-topon 22833  df-cn 23151
This theorem is referenced by:  cnmpt12f  23590  tgpmulg  24017  prdstgpd  24049  pcorevcl  24972  pcorevlem  24973  logcn  26601  loglesqrt  26713  efrlim  26921  efrlimOLD  26922  cvmliftlem8  34935  knoppcnlem10  36010  areacirclem2  37215  areacirclem4  37217
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