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Theorem cnmpt21f 22255
 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 21823 . . . 4 (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐿 ∈ Top)
64, 5syl 17 . . 3 (𝜑𝐿 ∈ Top)
7 toptopon2 21501 . . 3 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
86, 7sylib 221 . 2 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
9 eqid 2821 . . . . . 6 𝐿 = 𝐿
10 eqid 2821 . . . . . 6 𝑀 = 𝑀
119, 10cnf 21829 . . . . 5 (𝐹 ∈ (𝐿 Cn 𝑀) → 𝐹: 𝐿 𝑀)
124, 11syl 17 . . . 4 (𝜑𝐹: 𝐿 𝑀)
1312feqmptd 6706 . . 3 (𝜑𝐹 = (𝑧 𝐿 ↦ (𝐹𝑧)))
1413, 4eqeltrrd 2913 . 2 (𝜑 → (𝑧 𝐿 ↦ (𝐹𝑧)) ∈ (𝐿 Cn 𝑀))
15 fveq2 6643 . 2 (𝑧 = 𝐴 → (𝐹𝑧) = (𝐹𝐴))
161, 2, 3, 8, 14, 15cnmpt21 22254 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐹𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2115  ∪ cuni 4811   ↦ cmpt 5119  ⟶wf 6324  ‘cfv 6328  (class class class)co 7130   ∈ cmpo 7132  Topctop 21476  TopOnctopon 21493   Cn ccn 21807   ×t ctx 22143 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 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 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 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-map 8383  df-topgen 16695  df-top 21477  df-topon 21494  df-bases 21529  df-cn 21810  df-tx 22145 This theorem is referenced by:  cnmpt22  22257  cnmptk2  22269  txhmeo  22386  tgpsubcn  22673  istgp2  22674  dvrcn  22767  htpyid  23560  htpyco1  23561  reparphti  23580  pcocn  23600  pcorevlem  23609  cxpcn  25312  dipcn  28481  mndpluscn  31176  cvxsconn  32497  cvmlift2lem6  32562  cvmlift2lem12  32568
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