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Theorem cnmpt21f 21456
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 21025 . . . 4 (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐿 ∈ Top)
64, 5syl 17 . . 3 (𝜑𝐿 ∈ Top)
7 eqid 2620 . . . 4 𝐿 = 𝐿
87toptopon 20703 . . 3 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
96, 8sylib 208 . 2 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
10 eqid 2620 . . . . . 6 𝑀 = 𝑀
117, 10cnf 21031 . . . . 5 (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐹: 𝐿 𝑀)
124, 11syl 17 . . . 4 (𝜑𝐹: 𝐿 𝑀)
1312feqmptd 6236 . . 3 (𝜑𝐹 = (𝑧 𝐿 ↦ (𝐹𝑧)))
1413, 4eqeltrrd 2700 . 2 (𝜑 → (𝑧 𝐿 ↦ (𝐹𝑧)) ∈ (𝐿 Cn 𝑀))
15 fveq2 6178 . 2 (𝑧 = 𝐴 → (𝐹𝑧) = (𝐹𝐴))
161, 2, 3, 9, 14, 15cnmpt21 21455 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐹𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1988   cuni 4427  cmpt 4720  wf 5872  cfv 5876  (class class class)co 6635  cmpt2 6637  Topctop 20679  TopOnctopon 20696   Cn ccn 21009   ×t ctx 21344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-map 7844  df-topgen 16085  df-top 20680  df-topon 20697  df-bases 20731  df-cn 21012  df-tx 21346
This theorem is referenced by:  cnmpt22  21458  cnmptk2  21470  txhmeo  21587  tgpsubcn  21875  istgp2  21876  dvrcn  21968  htpyid  22757  htpyco1  22758  reparphti  22778  pcocn  22798  pcorevlem  22807  cxpcn  24467  dipcn  27545  mndpluscn  29946  cvxsconn  31199  cvmlift2lem6  31264  cvmlift2lem12  31270
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