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Theorem cnmpt21f 23175
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 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
cnmpt21.k (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
cnmpt21.a (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
cnmpt21f.f (πœ‘ β†’ 𝐹 ∈ (𝐿 Cn 𝑀))
Assertion
Ref Expression
cnmpt21f (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ (πΉβ€˜π΄)) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑀))
Distinct variable groups:   π‘₯,𝑦,𝐹   π‘₯,𝐿,𝑦   πœ‘,π‘₯,𝑦   π‘₯,𝑋,𝑦   π‘₯,𝑀,𝑦   π‘₯,π‘Œ,𝑦
Allowed substitution hints:   𝐴(π‘₯,𝑦)   𝐽(π‘₯,𝑦)   𝐾(π‘₯,𝑦)

Proof of Theorem cnmpt21f
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cnmpt21.j . 2 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2 cnmpt21.k . 2 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
3 cnmpt21.a . 2 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐴) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿))
4 cnmpt21f.f . . . 4 (πœ‘ β†’ 𝐹 ∈ (𝐿 Cn 𝑀))
5 cntop1 22743 . . . 4 (𝐹 ∈ (𝐿 Cn 𝑀) β†’ 𝐿 ∈ Top)
64, 5syl 17 . . 3 (πœ‘ β†’ 𝐿 ∈ Top)
7 toptopon2 22419 . . 3 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
86, 7sylib 217 . 2 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
9 eqid 2732 . . . . . 6 βˆͺ 𝐿 = βˆͺ 𝐿
10 eqid 2732 . . . . . 6 βˆͺ 𝑀 = βˆͺ 𝑀
119, 10cnf 22749 . . . . 5 (𝐹 ∈ (𝐿 Cn 𝑀) β†’ 𝐹:βˆͺ 𝐿⟢βˆͺ 𝑀)
124, 11syl 17 . . . 4 (πœ‘ β†’ 𝐹:βˆͺ 𝐿⟢βˆͺ 𝑀)
1312feqmptd 6960 . . 3 (πœ‘ β†’ 𝐹 = (𝑧 ∈ βˆͺ 𝐿 ↦ (πΉβ€˜π‘§)))
1413, 4eqeltrrd 2834 . 2 (πœ‘ β†’ (𝑧 ∈ βˆͺ 𝐿 ↦ (πΉβ€˜π‘§)) ∈ (𝐿 Cn 𝑀))
15 fveq2 6891 . 2 (𝑧 = 𝐴 β†’ (πΉβ€˜π‘§) = (πΉβ€˜π΄))
161, 2, 3, 8, 14, 15cnmpt21 23174 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ (πΉβ€˜π΄)) ∈ ((𝐽 Γ—t 𝐾) Cn 𝑀))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2106  βˆͺ cuni 4908   ↦ cmpt 5231  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   ∈ cmpo 7410  Topctop 22394  TopOnctopon 22411   Cn ccn 22727   Γ—t ctx 23063
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-map 8821  df-topgen 17388  df-top 22395  df-topon 22412  df-bases 22448  df-cn 22730  df-tx 23065
This theorem is referenced by:  cnmpt22  23177  cnmptk2  23189  txhmeo  23306  tgpsubcn  23593  istgp2  23594  dvrcn  23687  htpyid  24492  htpyco1  24493  reparphti  24512  pcocn  24532  pcorevlem  24541  cxpcn  26250  dipcn  29968  mndpluscn  32901  cvxsconn  34229  cvmlift2lem6  34294  cvmlift2lem12  34300  gg-reparphti  35167  gg-cxpcn  35179
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