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Theorem xkopjcn 23664
Description: Continuity of a projection map from the space of continuous functions. (This theorem can be strengthened, to joint continuity in both 𝑓 and 𝐴 as a function on (𝑆ko 𝑅) ×t 𝑅, but not without stronger assumptions on 𝑅; see xkofvcn 23692.) (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypothesis
Ref Expression
xkopjcn.1 𝑋 = 𝑅
Assertion
Ref Expression
xkopjcn ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝐴)) ∈ ((𝑆ko 𝑅) Cn 𝑆))
Distinct variable groups:   𝐴,𝑓   𝑅,𝑓   𝑆,𝑓   𝑓,𝑋

Proof of Theorem xkopjcn
StepHypRef Expression
1 eqid 2737 . . . . . 6 (𝑆ko 𝑅) = (𝑆ko 𝑅)
21xkotopon 23608 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
323adant3 1133 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
4 xkopjcn.1 . . . . . . . . 9 𝑋 = 𝑅
54topopn 22912 . . . . . . . 8 (𝑅 ∈ Top → 𝑋𝑅)
653ad2ant1 1134 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝑋𝑅)
7 fconst6g 6797 . . . . . . . 8 (𝑆 ∈ Top → (𝑋 × {𝑆}):𝑋⟶Top)
873ad2ant2 1135 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑋 × {𝑆}):𝑋⟶Top)
9 pttop 23590 . . . . . . 7 ((𝑋𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top) → (∏t‘(𝑋 × {𝑆})) ∈ Top)
106, 8, 9syl2anc 584 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (∏t‘(𝑋 × {𝑆})) ∈ Top)
11 eqid 2737 . . . . . . . . . 10 𝑆 = 𝑆
124, 11cnf 23254 . . . . . . . . 9 (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓:𝑋 𝑆)
13 uniexg 7760 . . . . . . . . . . 11 (𝑆 ∈ Top → 𝑆 ∈ V)
14133ad2ant2 1135 . . . . . . . . . 10 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝑆 ∈ V)
1514, 6elmapd 8880 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 ∈ ( 𝑆m 𝑋) ↔ 𝑓:𝑋 𝑆))
1612, 15imbitrrid 246 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓 ∈ ( 𝑆m 𝑋)))
1716ssrdv 3989 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑅 Cn 𝑆) ⊆ ( 𝑆m 𝑋))
18 simp2 1138 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝑆 ∈ Top)
19 eqid 2737 . . . . . . . . 9 (∏t‘(𝑋 × {𝑆})) = (∏t‘(𝑋 × {𝑆}))
2019, 11ptuniconst 23606 . . . . . . . 8 ((𝑋𝑅𝑆 ∈ Top) → ( 𝑆m 𝑋) = (∏t‘(𝑋 × {𝑆})))
216, 18, 20syl2anc 584 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ( 𝑆m 𝑋) = (∏t‘(𝑋 × {𝑆})))
2217, 21sseqtrd 4020 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑅 Cn 𝑆) ⊆ (∏t‘(𝑋 × {𝑆})))
23 eqid 2737 . . . . . . 7 (∏t‘(𝑋 × {𝑆})) = (∏t‘(𝑋 × {𝑆}))
2423restuni 23170 . . . . . 6 (((∏t‘(𝑋 × {𝑆})) ∈ Top ∧ (𝑅 Cn 𝑆) ⊆ (∏t‘(𝑋 × {𝑆}))) → (𝑅 Cn 𝑆) = ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)))
2510, 22, 24syl2anc 584 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑅 Cn 𝑆) = ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)))
2625fveq2d 6910 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (TopOn‘(𝑅 Cn 𝑆)) = (TopOn‘ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))))
273, 26eleqtrd 2843 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑆ko 𝑅) ∈ (TopOn‘ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))))
284, 19xkoptsub 23662 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆ko 𝑅))
29283adant3 1133 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆ko 𝑅))
30 eqid 2737 . . . 4 ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) = ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))
3130cnss1 23284 . . 3 (((𝑆ko 𝑅) ∈ (TopOn‘ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))) ∧ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆ko 𝑅)) → (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆) ⊆ ((𝑆ko 𝑅) Cn 𝑆))
3227, 29, 31syl2anc 584 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆) ⊆ ((𝑆ko 𝑅) Cn 𝑆))
3322resmptd 6058 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ↾ (𝑅 Cn 𝑆)) = (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝐴)))
34 simp3 1139 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝐴𝑋)
3523, 19ptpjcn 23619 . . . . . 6 ((𝑋𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top ∧ 𝐴𝑋) → (𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)))
366, 8, 34, 35syl3anc 1373 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)))
37 fvconst2g 7222 . . . . . . 7 ((𝑆 ∈ Top ∧ 𝐴𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆)
38373adant1 1131 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆)
3938oveq2d 7447 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)) = ((∏t‘(𝑋 × {𝑆})) Cn 𝑆))
4036, 39eleqtrd 2843 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn 𝑆))
4123cnrest 23293 . . . 4 (((𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn 𝑆) ∧ (𝑅 Cn 𝑆) ⊆ (∏t‘(𝑋 × {𝑆}))) → ((𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ↾ (𝑅 Cn 𝑆)) ∈ (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆))
4240, 22, 41syl2anc 584 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ↾ (𝑅 Cn 𝑆)) ∈ (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆))
4333, 42eqeltrrd 2842 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝐴)) ∈ (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆))
4432, 43sseldd 3984 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝐴)) ∈ ((𝑆ko 𝑅) Cn 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951  {csn 4626   cuni 4907  cmpt 5225   × cxp 5683  cres 5687  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  t crest 17465  tcpt 17483  Topctop 22899  TopOnctopon 22916   Cn ccn 23232  ko cxko 23569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-2o 8507  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-pt 17489  df-top 22900  df-topon 22917  df-bases 22953  df-cn 23235  df-cmp 23395  df-xko 23571
This theorem is referenced by:  cnmptkp  23688  xkofvcn  23692
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