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Theorem cnmpt11 23037
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 486 . . . . . . . . 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 22623 . . . . . . . . . . 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 1372 . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴):π‘‹βŸΆπ‘Œ)
76fvmptelcdm 7065 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ π‘Œ)
8 eqid 2733 . . . . . . . . . 10 (π‘₯ ∈ 𝑋 ↦ 𝐴) = (π‘₯ ∈ 𝑋 ↦ 𝐴)
98fvmpt2 6963 . . . . . . . . 9 ((π‘₯ ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯) = 𝐴)
101, 7, 9syl2anc 585 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯) = 𝐴)
1110fveq2d 6850 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡)β€˜((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯)) = ((𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π΄))
12 eqid 2733 . . . . . . . 8 (𝑦 ∈ π‘Œ ↦ 𝐡) = (𝑦 ∈ π‘Œ ↦ 𝐡)
13 cnmpt11.c . . . . . . . 8 (𝑦 = 𝐴 β†’ 𝐡 = 𝐢)
1413eleq1d 2819 . . . . . . . . 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 22615 . . . . . . . . . . . . . 14 ((𝑦 ∈ π‘Œ ↦ 𝐡) ∈ (𝐾 Cn 𝐿) β†’ 𝐿 ∈ Top)
1715, 16syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐿 ∈ Top)
18 toptopon2 22290 . . . . . . . . . . . . 13 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
1917, 18sylib 217 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜βˆͺ 𝐿))
20 cnf2 22623 . . . . . . . . . . . 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 1372 . . . . . . . . . . 11 (πœ‘ β†’ (𝑦 ∈ π‘Œ ↦ 𝐡):π‘ŒβŸΆβˆͺ 𝐿)
2212fmpt 7062 . . . . . . . . . . 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 482 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ βˆ€π‘¦ ∈ π‘Œ 𝐡 ∈ βˆͺ 𝐿)
2514, 24, 7rspcdva 3584 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐢 ∈ βˆͺ 𝐿)
2612, 13, 7, 25fvmptd3 6975 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡)β€˜π΄) = 𝐢)
2711, 26eqtrd 2773 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡)β€˜((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘₯)) = 𝐢)
28 fvco3 6944 . . . . . . 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 581 . . . . . 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 2733 . . . . . . . 8 (π‘₯ ∈ 𝑋 ↦ 𝐢) = (π‘₯ ∈ 𝑋 ↦ 𝐢)
3130fvmpt2 6963 . . . . . . 7 ((π‘₯ ∈ 𝑋 ∧ 𝐢 ∈ βˆͺ 𝐿) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯) = 𝐢)
321, 25, 31syl2anc 585 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯) = 𝐢)
3327, 29, 323eqtr4d 2783 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯))
3433ralrimiva 3140 . . . 4 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯))
35 nfv 1918 . . . . 5 Ⅎ𝑧(((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯)
36 nfcv 2904 . . . . . . . 8 β„²π‘₯(𝑦 ∈ π‘Œ ↦ 𝐡)
37 nfmpt1 5217 . . . . . . . 8 β„²π‘₯(π‘₯ ∈ 𝑋 ↦ 𝐴)
3836, 37nfco 5825 . . . . . . 7 β„²π‘₯((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))
39 nfcv 2904 . . . . . . 7 β„²π‘₯𝑧
4038, 39nffv 6856 . . . . . 6 β„²π‘₯(((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘§)
41 nfmpt1 5217 . . . . . . 7 β„²π‘₯(π‘₯ ∈ 𝑋 ↦ 𝐢)
4241, 39nffv 6856 . . . . . 6 β„²π‘₯((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘§)
4340, 42nfeq 2917 . . . . 5 β„²π‘₯(((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘§) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘§)
44 fveq2 6846 . . . . . 6 (π‘₯ = 𝑧 β†’ (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘₯) = (((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴))β€˜π‘§))
45 fveq2 6846 . . . . . 6 (π‘₯ = 𝑧 β†’ ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘₯) = ((π‘₯ ∈ 𝑋 ↦ 𝐢)β€˜π‘§))
4644, 45eqeq12d 2749 . . . . 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 3288 . . . 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 6696 . . . . . 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 585 . . . . 5 (πœ‘ β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)):π‘‹βŸΆβˆͺ 𝐿)
5150ffnd 6673 . . . 4 (πœ‘ β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)) Fn 𝑋)
5225fmpttd 7067 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐢):π‘‹βŸΆβˆͺ 𝐿)
5352ffnd 6673 . . . 4 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐢) Fn 𝑋)
54 eqfnfv 6986 . . . 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 585 . . 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 257 . 2 (πœ‘ β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)) = (π‘₯ ∈ 𝑋 ↦ 𝐢))
57 cnco 22640 . . 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 585 . 2 (πœ‘ β†’ ((𝑦 ∈ π‘Œ ↦ 𝐡) ∘ (π‘₯ ∈ 𝑋 ↦ 𝐴)) ∈ (𝐽 Cn 𝐿))
5956, 58eqeltrrd 2835 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐢) ∈ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆͺ cuni 4869   ↦ cmpt 5192   ∘ ccom 5641   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  Topctop 22265  TopOnctopon 22282   Cn ccn 22598
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-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-map 8773  df-top 22266  df-topon 22283  df-cn 22601
This theorem is referenced by:  cnmpt11f  23038  cnmptkp  23054  cnmptk1  23055  cnmpt1k  23056  ptunhmeo  23182  tmdgsum  23469  icchmeo  24327  evth2  24346  sinccvglem  34324  poimir  36161  broucube  36162
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