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Theorem cnmpt11f 23038
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 22615 . . . 4 ((π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) β†’ 𝐾 ∈ Top)
42, 3syl 17 . . 3 (πœ‘ β†’ 𝐾 ∈ Top)
5 toptopon2 22290 . . 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 2733 . . . . . 6 βˆͺ 𝐾 = βˆͺ 𝐾
9 eqid 2733 . . . . . 6 βˆͺ 𝐿 = βˆͺ 𝐿
108, 9cnf 22620 . . . . 5 (𝐹 ∈ (𝐾 Cn 𝐿) β†’ 𝐹:βˆͺ 𝐾⟢βˆͺ 𝐿)
117, 10syl 17 . . . 4 (πœ‘ β†’ 𝐹:βˆͺ 𝐾⟢βˆͺ 𝐿)
1211feqmptd 6914 . . 3 (πœ‘ β†’ 𝐹 = (𝑦 ∈ βˆͺ 𝐾 ↦ (πΉβ€˜π‘¦)))
1312, 7eqeltrrd 2835 . 2 (πœ‘ β†’ (𝑦 ∈ βˆͺ 𝐾 ↦ (πΉβ€˜π‘¦)) ∈ (𝐾 Cn 𝐿))
14 fveq2 6846 . 2 (𝑦 = 𝐴 β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π΄))
151, 2, 6, 13, 14cnmpt11 23037 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (πΉβ€˜π΄)) ∈ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2107  βˆͺ cuni 4869   ↦ cmpt 5192  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  Topctop 22265  TopOnctopon 22282   Cn ccn 22598
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-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-map 8773  df-top 22266  df-topon 22283  df-cn 22601
This theorem is referenced by:  cnmpt12f  23040  tgpmulg  23467  prdstgpd  23499  pcorevcl  24411  pcorevlem  24412  logcn  26025  loglesqrt  26134  efrlim  26342  cvmliftlem8  33950  knoppcnlem10  35018  areacirclem2  36217  areacirclem4  36219
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