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Theorem cnmpt21f 21696
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
cnmpt21.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmpt21.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmpt21.a (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
cnmpt21f.f (𝜑𝐹 ∈ (𝐿 Cn 𝑀))
Assertion
Ref Expression
cnmpt21f (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐹𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑀,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem cnmpt21f
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cnmpt21.j . 2 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 cnmpt21.k . 2 (𝜑𝐾 ∈ (TopOn‘𝑌))
3 cnmpt21.a . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
4 cnmpt21f.f . . . 4 (𝜑𝐹 ∈ (𝐿 Cn 𝑀))
5 cntop1 21265 . . . 4 (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐿 ∈ Top)
64, 5syl 17 . . 3 (𝜑𝐿 ∈ Top)
7 eqid 2771 . . . 4 𝐿 = 𝐿
87toptopon 20942 . . 3 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
96, 8sylib 208 . 2 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
10 eqid 2771 . . . . . 6 𝑀 = 𝑀
117, 10cnf 21271 . . . . 5 (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐹: 𝐿 𝑀)
124, 11syl 17 . . . 4 (𝜑𝐹: 𝐿 𝑀)
1312feqmptd 6393 . . 3 (𝜑𝐹 = (𝑧 𝐿 ↦ (𝐹𝑧)))
1413, 4eqeltrrd 2851 . 2 (𝜑 → (𝑧 𝐿 ↦ (𝐹𝑧)) ∈ (𝐿 Cn 𝑀))
15 fveq2 6333 . 2 (𝑧 = 𝐴 → (𝐹𝑧) = (𝐹𝐴))
161, 2, 3, 9, 14, 15cnmpt21 21695 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐹𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145   cuni 4575  cmpt 4864  wf 6026  cfv 6030  (class class class)co 6796  cmpt2 6798  Topctop 20918  TopOnctopon 20935   Cn ccn 21249   ×t ctx 21584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-map 8015  df-topgen 16312  df-top 20919  df-topon 20936  df-bases 20971  df-cn 21252  df-tx 21586
This theorem is referenced by:  cnmpt22  21698  cnmptk2  21710  txhmeo  21827  tgpsubcn  22114  istgp2  22115  dvrcn  22207  htpyid  22996  htpyco1  22997  reparphti  23016  pcocn  23036  pcorevlem  23045  cxpcn  24707  dipcn  27915  mndpluscn  30312  cvxsconn  31563  cvmlift2lem6  31628  cvmlift2lem12  31634
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