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Theorem xkopjcn 22356
 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 (𝑆 ↑ko 𝑅) ×t 𝑅, but not without stronger assumptions on 𝑅; see xkofvcn 22384.) (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 2758 . . . . . 6 (𝑆ko 𝑅) = (𝑆ko 𝑅)
21xkotopon 22300 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
323adant3 1129 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑆ko 𝑅) ∈ (TopOn‘(𝑅 Cn 𝑆)))
4 xkopjcn.1 . . . . . . . . 9 𝑋 = 𝑅
54topopn 21606 . . . . . . . 8 (𝑅 ∈ Top → 𝑋𝑅)
653ad2ant1 1130 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝑋𝑅)
7 fconst6g 6553 . . . . . . . 8 (𝑆 ∈ Top → (𝑋 × {𝑆}):𝑋⟶Top)
873ad2ant2 1131 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑋 × {𝑆}):𝑋⟶Top)
9 pttop 22282 . . . . . . 7 ((𝑋𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top) → (∏t‘(𝑋 × {𝑆})) ∈ Top)
106, 8, 9syl2anc 587 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (∏t‘(𝑋 × {𝑆})) ∈ Top)
11 eqid 2758 . . . . . . . . . 10 𝑆 = 𝑆
124, 11cnf 21946 . . . . . . . . 9 (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓:𝑋 𝑆)
13 uniexg 7464 . . . . . . . . . . 11 (𝑆 ∈ Top → 𝑆 ∈ V)
14133ad2ant2 1131 . . . . . . . . . 10 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝑆 ∈ V)
1514, 6elmapd 8430 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 ∈ ( 𝑆m 𝑋) ↔ 𝑓:𝑋 𝑆))
1612, 15syl5ibr 249 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) → 𝑓 ∈ ( 𝑆m 𝑋)))
1716ssrdv 3898 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑅 Cn 𝑆) ⊆ ( 𝑆m 𝑋))
18 simp2 1134 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝑆 ∈ Top)
19 eqid 2758 . . . . . . . . 9 (∏t‘(𝑋 × {𝑆})) = (∏t‘(𝑋 × {𝑆}))
2019, 11ptuniconst 22298 . . . . . . . 8 ((𝑋𝑅𝑆 ∈ Top) → ( 𝑆m 𝑋) = (∏t‘(𝑋 × {𝑆})))
216, 18, 20syl2anc 587 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ( 𝑆m 𝑋) = (∏t‘(𝑋 × {𝑆})))
2217, 21sseqtrd 3932 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑅 Cn 𝑆) ⊆ (∏t‘(𝑋 × {𝑆})))
23 eqid 2758 . . . . . . 7 (∏t‘(𝑋 × {𝑆})) = (∏t‘(𝑋 × {𝑆}))
2423restuni 21862 . . . . . 6 (((∏t‘(𝑋 × {𝑆})) ∈ Top ∧ (𝑅 Cn 𝑆) ⊆ (∏t‘(𝑋 × {𝑆}))) → (𝑅 Cn 𝑆) = ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)))
2510, 22, 24syl2anc 587 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑅 Cn 𝑆) = ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)))
2625fveq2d 6662 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (TopOn‘(𝑅 Cn 𝑆)) = (TopOn‘ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))))
273, 26eleqtrd 2854 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑆ko 𝑅) ∈ (TopOn‘ ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))))
284, 19xkoptsub 22354 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆ko 𝑅))
29283adant3 1129 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) ⊆ (𝑆ko 𝑅))
30 eqid 2758 . . . 4 ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) = ((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆))
3130cnss1 21976 . . 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 587 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆) ⊆ ((𝑆ko 𝑅) Cn 𝑆))
3322resmptd 5880 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ↾ (𝑅 Cn 𝑆)) = (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝐴)))
34 simp3 1135 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → 𝐴𝑋)
3523, 19ptpjcn 22311 . . . . . 6 ((𝑋𝑅 ∧ (𝑋 × {𝑆}):𝑋⟶Top ∧ 𝐴𝑋) → (𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)))
366, 8, 34, 35syl3anc 1368 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)))
37 fvconst2g 6955 . . . . . . 7 ((𝑆 ∈ Top ∧ 𝐴𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆)
38373adant1 1127 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((𝑋 × {𝑆})‘𝐴) = 𝑆)
3938oveq2d 7166 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → ((∏t‘(𝑋 × {𝑆})) Cn ((𝑋 × {𝑆})‘𝐴)) = ((∏t‘(𝑋 × {𝑆})) Cn 𝑆))
4036, 39eleqtrd 2854 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 (∏t‘(𝑋 × {𝑆})) ↦ (𝑓𝐴)) ∈ ((∏t‘(𝑋 × {𝑆})) Cn 𝑆))
4123cnrest 21985 . . . 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 587 . . 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 2853 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝐴)) ∈ (((∏t‘(𝑋 × {𝑆})) ↾t (𝑅 Cn 𝑆)) Cn 𝑆))
4432, 43sseldd 3893 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 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ⊆ wss 3858  {csn 4522  ∪ cuni 4798   ↦ cmpt 5112   × cxp 5522   ↾ cres 5526  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150   ↑m cmap 8416   ↾t crest 16752  ∏tcpt 16770  Topctop 21593  TopOnctopon 21610   Cn ccn 21924   ↑ko cxko 22261 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-iin 4886  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-1o 8112  df-er 8299  df-map 8418  df-ixp 8480  df-en 8528  df-dom 8529  df-fin 8531  df-fi 8908  df-rest 16754  df-topgen 16775  df-pt 16776  df-top 21594  df-topon 21611  df-bases 21646  df-cn 21927  df-cmp 22087  df-xko 22263 This theorem is referenced by:  cnmptkp  22380  xkofvcn  22384
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