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Theorem cnmpt11 23166
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))
cnmpt11.k (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
cnmpt11.b (πœ‘ β†’ (𝑦 ∈ π‘Œ ↦ 𝐡) ∈ (𝐾 Cn 𝐿))
cnmpt11.c (𝑦 = 𝐴 β†’ 𝐡 = 𝐢)
Assertion
Ref Expression
cnmpt11 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐢) ∈ (𝐽 Cn 𝐿))
Distinct variable groups:   𝑦,𝐴   π‘₯,𝑦   πœ‘,π‘₯   π‘₯,𝐽,𝑦   π‘₯,𝑋,𝑦   π‘₯,π‘Œ,𝑦   π‘₯,𝐾,𝑦   π‘₯,𝐿,𝑦   π‘₯,𝐡   𝑦,𝐢
Allowed substitution hints:   πœ‘(𝑦)   𝐴(π‘₯)   𝐡(𝑦)   𝐢(π‘₯)

Proof of Theorem cnmpt11
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 simpr 485 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
2 cnmptid.j . . . . . . . . . . 11 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
3 cnmpt11.k . . . . . . . . . . 11 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
4 cnmpt11.a . . . . . . . . . . 11 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾))
5 cnf2 22752 . . . . . . . . . . 11 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴):π‘‹βŸΆπ‘Œ)
62, 3, 4, 5syl3anc 1371 . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴):π‘‹βŸΆπ‘Œ)
76fvmptelcdm 7112 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ π‘Œ)
8 eqid 2732 . . . . . . . . . 10 (π‘₯ ∈ 𝑋 ↦ 𝐴) = (π‘₯ ∈ 𝑋 ↦ 𝐴)
98fvmpt2 7009 . . . . . . . . 9 ((π‘₯ ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯) = 𝐴)
101, 7, 9syl2anc 584 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯) = 𝐴)
1110fveq2d 6895 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡)β€˜((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯)) = ((𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π΄))
12 eqid 2732 . . . . . . . 8 (𝑦 ∈ π‘Œ ↦ 𝐡) = (𝑦 ∈ π‘Œ ↦ 𝐡)
13 cnmpt11.c . . . . . . . 8 (𝑦 = 𝐴 β†’ 𝐡 = 𝐢)
1413eleq1d 2818 . . . . . . . . 9 (𝑦 = 𝐴 β†’ (𝐡 ∈ βˆͺ 𝐿 ↔ 𝐢 ∈ βˆͺ 𝐿))
15 cnmpt11.b . . . . . . . . . . . . . 14 (πœ‘ β†’ (𝑦 ∈ π‘Œ ↦ 𝐡) ∈ (𝐾 Cn 𝐿))
16 cntop2 22744 . . . . . . . . . . . . . 14 ((𝑦 ∈ π‘Œ ↦ 𝐡) ∈ (𝐾 Cn 𝐿) β†’ 𝐿 ∈ Top)
1715, 16syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐿 ∈ Top)
18 toptopon2 22419 . . . . . . . . . . . . 13 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
1917, 18sylib 217 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
20 cnf2 22752 . . . . . . . . . . . 12 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿) ∧ (𝑦 ∈ π‘Œ ↦ 𝐡) ∈ (𝐾 Cn 𝐿)) β†’ (𝑦 ∈ π‘Œ ↦ 𝐡):π‘ŒβŸΆβˆͺ 𝐿)
213, 19, 15, 20syl3anc 1371 . . . . . . . . . . 11 (πœ‘ β†’ (𝑦 ∈ π‘Œ ↦ 𝐡):π‘ŒβŸΆβˆͺ 𝐿)
2212fmpt 7109 . . . . . . . . . . 11 (βˆ€π‘¦ ∈ π‘Œ 𝐡 ∈ βˆͺ 𝐿 ↔ (𝑦 ∈ π‘Œ ↦ 𝐡):π‘ŒβŸΆβˆͺ 𝐿)
2321, 22sylibr 233 . . . . . . . . . 10 (πœ‘ β†’ βˆ€π‘¦ ∈ π‘Œ 𝐡 ∈ βˆͺ 𝐿)
2423adantr 481 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ βˆ€π‘¦ ∈ π‘Œ 𝐡 ∈ βˆͺ 𝐿)
2514, 24, 7rspcdva 3613 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐢 ∈ βˆͺ 𝐿)
2612, 13, 7, 25fvmptd3 7021 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π΄) = 𝐢)
2711, 26eqtrd 2772 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡)β€˜((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯)) = 𝐢)
28 fvco3 6990 . . . . . . 7 (((π‘₯ ∈ 𝑋 ↦ 𝐴):π‘‹βŸΆπ‘Œ ∧ π‘₯ ∈ 𝑋) β†’ (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = ((𝑦 ∈ π‘Œ ↦ 𝐡)β€˜((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯)))
296, 28sylan 580 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = ((𝑦 ∈ π‘Œ ↦ 𝐡)β€˜((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯)))
30 eqid 2732 . . . . . . . 8 (π‘₯ ∈ 𝑋 ↦ 𝐢) = (π‘₯ ∈ 𝑋 ↦ 𝐢)
3130fvmpt2 7009 . . . . . . 7 ((π‘₯ ∈ 𝑋 ∧ 𝐢 ∈ βˆͺ 𝐿) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯) = 𝐢)
321, 25, 31syl2anc 584 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯) = 𝐢)
3327, 29, 323eqtr4d 2782 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯))
3433ralrimiva 3146 . . . 4 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯))
35 nfv 1917 . . . . 5 Ⅎ𝑧(((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯)
36 nfcv 2903 . . . . . . . 8 β„²π‘₯(𝑦 ∈ π‘Œ ↦ 𝐡)
37 nfmpt1 5256 . . . . . . . 8 β„²π‘₯(π‘₯ ∈ 𝑋 ↦ 𝐴)
3836, 37nfco 5865 . . . . . . 7 β„²π‘₯((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))
39 nfcv 2903 . . . . . . 7 β„²π‘₯𝑧
4038, 39nffv 6901 . . . . . 6 β„²π‘₯(((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘§)
41 nfmpt1 5256 . . . . . . 7 β„²π‘₯(π‘₯ ∈ 𝑋 ↦ 𝐢)
4241, 39nffv 6901 . . . . . 6 β„²π‘₯((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘§)
4340, 42nfeq 2916 . . . . 5 β„²π‘₯(((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘§) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘§)
44 fveq2 6891 . . . . . 6 (π‘₯ = 𝑧 β†’ (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘§))
45 fveq2 6891 . . . . . 6 (π‘₯ = 𝑧 β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘§))
4644, 45eqeq12d 2748 . . . . 5 (π‘₯ = 𝑧 β†’ ((((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯) ↔ (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘§) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘§)))
4735, 43, 46cbvralw 3303 . . . 4 (βˆ€π‘₯ ∈ 𝑋 (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯) ↔ βˆ€π‘§ ∈ 𝑋 (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘§) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘§))
4834, 47sylib 217 . . 3 (πœ‘ β†’ βˆ€π‘§ ∈ 𝑋 (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘§) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘§))
49 fco 6741 . . . . . 6 (((𝑦 ∈ π‘Œ ↦ 𝐡):π‘ŒβŸΆβˆͺ 𝐿 ∧ (π‘₯ ∈ 𝑋 ↦ 𝐴):π‘‹βŸΆπ‘Œ) β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)):π‘‹βŸΆβˆͺ 𝐿)
5021, 6, 49syl2anc 584 . . . . 5 (πœ‘ β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)):π‘‹βŸΆβˆͺ 𝐿)
5150ffnd 6718 . . . 4 (πœ‘ β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)) Fn 𝑋)
5225fmpttd 7114 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐢):π‘‹βŸΆβˆͺ 𝐿)
5352ffnd 6718 . . . 4 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐢) Fn 𝑋)
54 eqfnfv 7032 . . . 4 ((((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)) Fn 𝑋 ∧ (π‘₯ ∈ 𝑋 ↦ 𝐢) Fn 𝑋) β†’ (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)) = (π‘₯ ∈ 𝑋 ↦ 𝐢) ↔ βˆ€π‘§ ∈ 𝑋 (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘§) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘§)))
5551, 53, 54syl2anc 584 . . 3 (πœ‘ β†’ (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)) = (π‘₯ ∈ 𝑋 ↦ 𝐢) ↔ βˆ€π‘§ ∈ 𝑋 (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘§) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘§)))
5648, 55mpbird 256 . 2 (πœ‘ β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)) = (π‘₯ ∈ 𝑋 ↦ 𝐢))
57 cnco 22769 . . 3 (((π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾) ∧ (𝑦 ∈ π‘Œ ↦ 𝐡) ∈ (𝐾 Cn 𝐿)) β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)) ∈ (𝐽 Cn 𝐿))
584, 15, 57syl2anc 584 . 2 (πœ‘ β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)) ∈ (𝐽 Cn 𝐿))
5956, 58eqeltrrd 2834 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐢) ∈ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆͺ cuni 4908   ↦ cmpt 5231   ∘ ccom 5680   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  Topctop 22394  TopOnctopon 22411   Cn ccn 22727
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-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-map 8821  df-top 22395  df-topon 22412  df-cn 22730
This theorem is referenced by:  cnmpt11f  23167  cnmptkp  23183  cnmptk1  23184  cnmpt1k  23185  ptunhmeo  23311  tmdgsum  23598  icchmeo  24456  evth2  24475  sinccvglem  34652  gg-icchmeo  35165  poimir  36516  broucube  36517
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