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Theorem cnmpt12f 22274
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))
cnmpt1t.b (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))
cnmpt12f.f (𝜑𝐹 ∈ ((𝐾 ×t 𝐿) Cn 𝑀))
Assertion
Ref Expression
cnmpt12f (𝜑 → (𝑥𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝐽 Cn 𝑀))
Distinct variable groups:   𝑥,𝐹   𝜑,𝑥   𝑥,𝐽   𝑥,𝑀   𝑥,𝑋   𝑥,𝐾   𝑥,𝐿
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem cnmpt12f
StepHypRef Expression
1 df-ov 7152 . . 3 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
21mpteq2i 5144 . 2 (𝑥𝑋 ↦ (𝐴𝐹𝐵)) = (𝑥𝑋 ↦ (𝐹‘⟨𝐴, 𝐵⟩))
3 cnmptid.j . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
4 cnmpt11.a . . . 4 (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))
5 cnmpt1t.b . . . 4 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))
63, 4, 5cnmpt1t 22273 . . 3 (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))
7 cnmpt12f.f . . 3 (𝜑𝐹 ∈ ((𝐾 ×t 𝐿) Cn 𝑀))
83, 6, 7cnmpt11f 22272 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐹‘⟨𝐴, 𝐵⟩)) ∈ (𝐽 Cn 𝑀))
92, 8eqeltrid 2920 1 (𝜑 → (𝑥𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝐽 Cn 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  cop 4556  cmpt 5132  cfv 6343  (class class class)co 7149  TopOnctopon 21518   Cn ccn 21832   ×t ctx 22168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-map 8404  df-topgen 16717  df-top 21502  df-topon 21519  df-bases 21554  df-cn 21835  df-tx 22170
This theorem is referenced by:  cnmpt12  22275  cnmpt1plusg  22695  istgp2  22699  clsnsg  22718  tgpt0  22727  cnmpt1vsca  22802  cnmpt1ds  23450  fsumcn  23478  expcn  23480  divccn  23481  cncfmpt2f  23523  cdivcncf  23529  iirevcn  23538  iihalf1cn  23540  iihalf2cn  23542  icchmeo  23549  evth  23567  evth2  23568  pcoass  23632  cnmpt1ip  23854  dvcnvlem  24582  plycn  24861  psercn2  25021  atansopn  25521  efrlim  25558  ipasslem7  28622  occllem  29089  hmopidmchi  29937  cvxpconn  32546  cvmlift2lem2  32608  cvmlift2lem3  32609  cvmliftphtlem  32621  sinccvglem  32972  knoppcnlem10  33898  broucube  35036  areacirclem2  35091  fprodcnlem  42167
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