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Theorem cnmpt1t 13788
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
cnmptid.j (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
cnmpt11.a (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
cnmpt1t.b (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐿))
Assertion
Ref Expression
cnmpt1t (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ ⟨𝐴, 𝐡⟩) ∈ (𝐽 Cn (𝐾 Γ—t 𝐿)))
Distinct variable groups:   πœ‘,π‘₯   π‘₯,𝐽   π‘₯,𝑋   π‘₯,𝐾   π‘₯,𝐿
Allowed substitution hints:   𝐴(π‘₯)   𝐡(π‘₯)

Proof of Theorem cnmpt1t
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cnmptid.j . . . 4 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2 toponuni 13518 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
3 mpteq1 4088 . . . 4 (𝑋 = βˆͺ 𝐽 β†’ (π‘₯ ∈ 𝑋 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩) = (π‘₯ ∈ βˆͺ 𝐽 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩))
41, 2, 33syl 17 . . 3 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩) = (π‘₯ ∈ βˆͺ 𝐽 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩))
5 simpr 110 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
6 cnmpt11.a . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
7 cntop2 13705 . . . . . . . . . 10 ((π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) β†’ 𝐾 ∈ Top)
86, 7syl 14 . . . . . . . . 9 (πœ‘ β†’ 𝐾 ∈ Top)
9 toptopon2 13522 . . . . . . . . 9 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
108, 9sylib 122 . . . . . . . 8 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
11 cnf2 13708 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) ∧ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴):π‘‹βŸΆβˆͺ 𝐾)
121, 10, 6, 11syl3anc 1238 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴):π‘‹βŸΆβˆͺ 𝐾)
1312fvmptelcdm 5670 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ βˆͺ 𝐾)
14 eqid 2177 . . . . . . 7 (π‘₯ ∈ 𝑋 ↦ 𝐴) = (π‘₯ ∈ 𝑋 ↦ 𝐴)
1514fvmpt2 5600 . . . . . 6 ((π‘₯ ∈ 𝑋 ∧ 𝐴 ∈ βˆͺ 𝐾) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯) = 𝐴)
165, 13, 15syl2anc 411 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯) = 𝐴)
17 cnmpt1t.b . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐿))
18 cntop2 13705 . . . . . . . . . 10 ((π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐿) β†’ 𝐿 ∈ Top)
1917, 18syl 14 . . . . . . . . 9 (πœ‘ β†’ 𝐿 ∈ Top)
20 toptopon2 13522 . . . . . . . . 9 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
2119, 20sylib 122 . . . . . . . 8 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
22 cnf2 13708 . . . . . . . 8 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿) ∧ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐿)) β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡):π‘‹βŸΆβˆͺ 𝐿)
231, 21, 17, 22syl3anc 1238 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡):π‘‹βŸΆβˆͺ 𝐿)
2423fvmptelcdm 5670 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐡 ∈ βˆͺ 𝐿)
25 eqid 2177 . . . . . . 7 (π‘₯ ∈ 𝑋 ↦ 𝐡) = (π‘₯ ∈ 𝑋 ↦ 𝐡)
2625fvmpt2 5600 . . . . . 6 ((π‘₯ ∈ 𝑋 ∧ 𝐡 ∈ βˆͺ 𝐿) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯) = 𝐡)
275, 24, 26syl2anc 411 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯) = 𝐡)
2816, 27opeq12d 3787 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩ = ⟨𝐴, 𝐡⟩)
2928mpteq2dva 4094 . . 3 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩) = (π‘₯ ∈ 𝑋 ↦ ⟨𝐴, 𝐡⟩))
304, 29eqtr3d 2212 . 2 (πœ‘ β†’ (π‘₯ ∈ βˆͺ 𝐽 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩) = (π‘₯ ∈ 𝑋 ↦ ⟨𝐴, 𝐡⟩))
31 eqid 2177 . . . 4 βˆͺ 𝐽 = βˆͺ 𝐽
32 nfcv 2319 . . . . 5 β„²π‘¦βŸ¨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩
33 nffvmpt1 5527 . . . . . 6 β„²π‘₯((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘¦)
34 nffvmpt1 5527 . . . . . 6 β„²π‘₯((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘¦)
3533, 34nfop 3795 . . . . 5 β„²π‘₯⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘¦), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘¦)⟩
36 fveq2 5516 . . . . . 6 (π‘₯ = 𝑦 β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘¦))
37 fveq2 5516 . . . . . 6 (π‘₯ = 𝑦 β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘¦))
3836, 37opeq12d 3787 . . . . 5 (π‘₯ = 𝑦 β†’ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩ = ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘¦), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘¦)⟩)
3932, 35, 38cbvmpt 4099 . . . 4 (π‘₯ ∈ βˆͺ 𝐽 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩) = (𝑦 ∈ βˆͺ 𝐽 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘¦), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘¦)⟩)
4031, 39txcnmpt 13776 . . 3 (((π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) ∧ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐿)) β†’ (π‘₯ ∈ βˆͺ 𝐽 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩) ∈ (𝐽 Cn (𝐾 Γ—t 𝐿)))
416, 17, 40syl2anc 411 . 2 (πœ‘ β†’ (π‘₯ ∈ βˆͺ 𝐽 ↦ ⟨((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯), ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘₯)⟩) ∈ (𝐽 Cn (𝐾 Γ—t 𝐿)))
4230, 41eqeltrrd 2255 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ ⟨𝐴, 𝐡⟩) ∈ (𝐽 Cn (𝐾 Γ—t 𝐿)))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  βŸ¨cop 3596  βˆͺ cuni 3810   ↦ cmpt 4065  βŸΆwf 5213  β€˜cfv 5217  (class class class)co 5875  Topctop 13500  TopOnctopon 13513   Cn ccn 13688   Γ—t ctx 13755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-map 6650  df-topgen 12709  df-top 13501  df-topon 13514  df-bases 13546  df-cn 13691  df-tx 13756
This theorem is referenced by:  cnmpt12f  13789  imasnopn  13802  cnrehmeocntop  14096
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