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Theorem cnmpt11f 22271
 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 21848 . . . 4 ((𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
42, 3syl 17 . . 3 (𝜑𝐾 ∈ Top)
5 toptopon2 21525 . . 3 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
64, 5sylib 220 . 2 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
7 cnmpt11f.f . . . . 5 (𝜑𝐹 ∈ (𝐾 Cn 𝐿))
8 eqid 2821 . . . . . 6 𝐾 = 𝐾
9 eqid 2821 . . . . . 6 𝐿 = 𝐿
108, 9cnf 21853 . . . . 5 (𝐹 ∈ (𝐾 Cn 𝐿) → 𝐹: 𝐾 𝐿)
117, 10syl 17 . . . 4 (𝜑𝐹: 𝐾 𝐿)
1211feqmptd 6732 . . 3 (𝜑𝐹 = (𝑦 𝐾 ↦ (𝐹𝑦)))
1312, 7eqeltrrd 2914 . 2 (𝜑 → (𝑦 𝐾 ↦ (𝐹𝑦)) ∈ (𝐾 Cn 𝐿))
14 fveq2 6669 . 2 (𝑦 = 𝐴 → (𝐹𝑦) = (𝐹𝐴))
151, 2, 6, 13, 14cnmpt11 22270 1 (𝜑 → (𝑥𝑋 ↦ (𝐹𝐴)) ∈ (𝐽 Cn 𝐿))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2110  ∪ cuni 4837   ↦ cmpt 5145  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155  Topctop 21500  TopOnctopon 21517   Cn ccn 21831 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-map 8407  df-top 21501  df-topon 21518  df-cn 21834 This theorem is referenced by:  cnmpt12f  22273  tgpmulg  22700  prdstgpd  22732  pcorevcl  23628  pcorevlem  23629  logcn  25229  loglesqrt  25338  efrlim  25546  cvmliftlem8  32539  knoppcnlem10  33841  areacirclem2  34982  areacirclem4  34984
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