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Theorem cnmpt11f 23612
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 23189 . . . 4 ((𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
42, 3syl 17 . . 3 (𝜑𝐾 ∈ Top)
5 toptopon2 22864 . . 3 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
64, 5sylib 217 . 2 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
7 cnmpt11f.f . . . . 5 (𝜑𝐹 ∈ (𝐾 Cn 𝐿))
8 eqid 2725 . . . . . 6 𝐾 = 𝐾
9 eqid 2725 . . . . . 6 𝐿 = 𝐿
108, 9cnf 23194 . . . . 5 (𝐹 ∈ (𝐾 Cn 𝐿) → 𝐹: 𝐾 𝐿)
117, 10syl 17 . . . 4 (𝜑𝐹: 𝐾 𝐿)
1211feqmptd 6966 . . 3 (𝜑𝐹 = (𝑦 𝐾 ↦ (𝐹𝑦)))
1312, 7eqeltrrd 2826 . 2 (𝜑 → (𝑦 𝐾 ↦ (𝐹𝑦)) ∈ (𝐾 Cn 𝐿))
14 fveq2 6896 . 2 (𝑦 = 𝐴 → (𝐹𝑦) = (𝐹𝐴))
151, 2, 6, 13, 14cnmpt11 23611 1 (𝜑 → (𝑥𝑋 ↦ (𝐹𝐴)) ∈ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098   cuni 4909  cmpt 5232  wf 6545  cfv 6549  (class class class)co 7419  Topctop 22839  TopOnctopon 22856   Cn ccn 23172
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 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8847  df-top 22840  df-topon 22857  df-cn 23175
This theorem is referenced by:  cnmpt12f  23614  tgpmulg  24041  prdstgpd  24073  pcorevcl  24996  pcorevlem  24997  logcn  26626  loglesqrt  26738  efrlim  26946  efrlimOLD  26947  cvmliftlem8  35033  knoppcnlem10  36108  areacirclem2  37313  areacirclem4  37315
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