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Theorem cnmpt11f 21515
 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))
cnmpt11f.f (𝜑𝐹 ∈ (𝐾 Cn 𝐿))
Assertion
Ref Expression
cnmpt11f (𝜑 → (𝑥𝑋 ↦ (𝐹𝐴)) ∈ (𝐽 Cn 𝐿))
Distinct variable groups:   𝑥,𝐹   𝜑,𝑥   𝑥,𝐽   𝑥,𝑋   𝑥,𝐾   𝑥,𝐿
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem cnmpt11f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cnmptid.j . 2 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 cnmpt11.a . 2 (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))
3 cntop2 21093 . . . 4 ((𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
42, 3syl 17 . . 3 (𝜑𝐾 ∈ Top)
5 eqid 2651 . . . 4 𝐾 = 𝐾
65toptopon 20770 . . 3 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
74, 6sylib 208 . 2 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
8 cnmpt11f.f . . . . 5 (𝜑𝐹 ∈ (𝐾 Cn 𝐿))
9 eqid 2651 . . . . . 6 𝐿 = 𝐿
105, 9cnf 21098 . . . . 5 (𝐹 ∈ (𝐾 Cn 𝐿) → 𝐹: 𝐾 𝐿)
118, 10syl 17 . . . 4 (𝜑𝐹: 𝐾 𝐿)
1211feqmptd 6288 . . 3 (𝜑𝐹 = (𝑦 𝐾 ↦ (𝐹𝑦)))
1312, 8eqeltrrd 2731 . 2 (𝜑 → (𝑦 𝐾 ↦ (𝐹𝑦)) ∈ (𝐾 Cn 𝐿))
14 fveq2 6229 . 2 (𝑦 = 𝐴 → (𝐹𝑦) = (𝐹𝐴))
151, 2, 7, 13, 14cnmpt11 21514 1 (𝜑 → (𝑥𝑋 ↦ (𝐹𝐴)) ∈ (𝐽 Cn 𝐿))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2030  ∪ cuni 4468   ↦ cmpt 4762  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  Topctop 20746  TopOnctopon 20763   Cn ccn 21076 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-top 20747  df-topon 20764  df-cn 21079 This theorem is referenced by:  cnmpt12f  21517  tgpmulg  21944  prdstgpd  21975  pcorevcl  22871  pcorevlem  22872  logcn  24438  loglesqrt  24544  efrlim  24741  cvmliftlem8  31400  knoppcnlem10  32617  areacirclem2  33631  areacirclem4  33633
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