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Theorem cnmpt12f 23033
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 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
cnmpt11.a (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
cnmpt1t.b (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐿))
cnmpt12f.f (πœ‘ β†’ 𝐹 ∈ ((𝐾 Γ—t 𝐿) Cn 𝑀))
Assertion
Ref Expression
cnmpt12f (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴𝐹𝐡)) ∈ (𝐽 Cn 𝑀))
Distinct variable groups:   π‘₯,𝐹   πœ‘,π‘₯   π‘₯,𝐽   π‘₯,𝑀   π‘₯,𝑋   π‘₯,𝐾   π‘₯,𝐿
Allowed substitution hints:   𝐴(π‘₯)   𝐡(π‘₯)

Proof of Theorem cnmpt12f
StepHypRef Expression
1 df-ov 7361 . . 3 (𝐴𝐹𝐡) = (πΉβ€˜βŸ¨π΄, 𝐡⟩)
21mpteq2i 5211 . 2 (π‘₯ ∈ 𝑋 ↦ (𝐴𝐹𝐡)) = (π‘₯ ∈ 𝑋 ↦ (πΉβ€˜βŸ¨π΄, 𝐡⟩))
3 cnmptid.j . . 3 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
4 cnmpt11.a . . . 4 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
5 cnmpt1t.b . . . 4 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐿))
63, 4, 5cnmpt1t 23032 . . 3 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ ⟨𝐴, 𝐡⟩) ∈ (𝐽 Cn (𝐾 Γ—t 𝐿)))
7 cnmpt12f.f . . 3 (πœ‘ β†’ 𝐹 ∈ ((𝐾 Γ—t 𝐿) Cn 𝑀))
83, 6, 7cnmpt11f 23031 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (πΉβ€˜βŸ¨π΄, 𝐡⟩)) ∈ (𝐽 Cn 𝑀))
92, 8eqeltrid 2838 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴𝐹𝐡)) ∈ (𝐽 Cn 𝑀))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2107  βŸ¨cop 4593   ↦ cmpt 5189  β€˜cfv 6497  (class class class)co 7358  TopOnctopon 22275   Cn ccn 22591   Γ—t ctx 22927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8770  df-topgen 17330  df-top 22259  df-topon 22276  df-bases 22312  df-cn 22594  df-tx 22929
This theorem is referenced by:  cnmpt12  23034  cnmpt1plusg  23454  istgp2  23458  clsnsg  23477  tgpt0  23486  cnmpt1vsca  23561  cnmpt1ds  24221  fsumcn  24249  expcn  24251  divccn  24252  cncfmpt2f  24294  cdivcncf  24300  iirevcn  24309  iihalf1cn  24311  iihalf2cn  24313  icchmeo  24320  evth  24338  evth2  24339  pcoass  24403  cnmpt1ip  24627  dvcnvlem  25356  plycn  25638  psercn2  25798  atansopn  26298  efrlim  26335  ipasslem7  29820  occllem  30287  hmopidmchi  31135  cvxpconn  33893  cvmlift2lem2  33955  cvmlift2lem3  33956  cvmliftphtlem  33968  sinccvglem  34317  knoppcnlem10  35011  broucube  36158  areacirclem2  36213  fprodcnlem  43926
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