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Theorem cnmpt11f 21845
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 21423 . . . 4 ((𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
42, 3syl 17 . . 3 (𝜑𝐾 ∈ Top)
5 eqid 2825 . . . 4 𝐾 = 𝐾
65toptopon 21099 . . 3 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
74, 6sylib 210 . 2 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
8 cnmpt11f.f . . . . 5 (𝜑𝐹 ∈ (𝐾 Cn 𝐿))
9 eqid 2825 . . . . . 6 𝐿 = 𝐿
105, 9cnf 21428 . . . . 5 (𝐹 ∈ (𝐾 Cn 𝐿) → 𝐹: 𝐾 𝐿)
118, 10syl 17 . . . 4 (𝜑𝐹: 𝐾 𝐿)
1211feqmptd 6500 . . 3 (𝜑𝐹 = (𝑦 𝐾 ↦ (𝐹𝑦)))
1312, 8eqeltrrd 2907 . 2 (𝜑 → (𝑦 𝐾 ↦ (𝐹𝑦)) ∈ (𝐾 Cn 𝐿))
14 fveq2 6437 . 2 (𝑦 = 𝐴 → (𝐹𝑦) = (𝐹𝐴))
151, 2, 7, 13, 14cnmpt11 21844 1 (𝜑 → (𝑥𝑋 ↦ (𝐹𝐴)) ∈ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2164   cuni 4660  cmpt 4954  wf 6123  cfv 6127  (class class class)co 6910  Topctop 21075  TopOnctopon 21092   Cn ccn 21406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-map 8129  df-top 21076  df-topon 21093  df-cn 21409
This theorem is referenced by:  cnmpt12f  21847  tgpmulg  22274  prdstgpd  22305  pcorevcl  23201  pcorevlem  23202  logcn  24799  loglesqrt  24908  efrlim  25116  cvmliftlem8  31816  knoppcnlem10  33020  areacirclem2  34039  areacirclem4  34041
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