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Theorem cnmpt21f 23046
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 22614 . . . 4 (𝐹 ∈ (𝐿 Cn 𝑀) β†’ 𝐿 ∈ Top)
64, 5syl 17 . . 3 (πœ‘ β†’ 𝐿 ∈ Top)
7 toptopon2 22290 . . 3 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
86, 7sylib 217 . 2 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
9 eqid 2733 . . . . . 6 βˆͺ 𝐿 = βˆͺ 𝐿
10 eqid 2733 . . . . . 6 βˆͺ 𝑀 = βˆͺ 𝑀
119, 10cnf 22620 . . . . 5 (𝐹 ∈ (𝐿 Cn 𝑀) β†’ 𝐹:βˆͺ 𝐿⟢βˆͺ 𝑀)
124, 11syl 17 . . . 4 (πœ‘ β†’ 𝐹:βˆͺ 𝐿⟢βˆͺ 𝑀)
1312feqmptd 6914 . . 3 (πœ‘ β†’ 𝐹 = (𝑧 ∈ βˆͺ 𝐿 ↦ (πΉβ€˜π‘§)))
1413, 4eqeltrrd 2835 . 2 (πœ‘ β†’ (𝑧 ∈ βˆͺ 𝐿 ↦ (πΉβ€˜π‘§)) ∈ (𝐿 Cn 𝑀))
15 fveq2 6846 . 2 (𝑧 = 𝐴 β†’ (πΉβ€˜π‘§) = (πΉβ€˜π΄))
161, 2, 3, 8, 14, 15cnmpt21 23045 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 2107  βˆͺ cuni 4869   ↦ cmpt 5192  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  Topctop 22265  TopOnctopon 22282   Cn ccn 22598   Γ—t ctx 22934
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 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-map 8773  df-topgen 17333  df-top 22266  df-topon 22283  df-bases 22319  df-cn 22601  df-tx 22936
This theorem is referenced by:  cnmpt22  23048  cnmptk2  23060  txhmeo  23177  tgpsubcn  23464  istgp2  23465  dvrcn  23558  htpyid  24363  htpyco1  24364  reparphti  24383  pcocn  24403  pcorevlem  24412  cxpcn  26121  dipcn  29711  mndpluscn  32571  cvxsconn  33901  cvmlift2lem6  33966  cvmlift2lem12  33972
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