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Theorem ptpjcn 22147
Description: Continuity of a projection map into a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Feb-2015.)
Hypotheses
Ref Expression
ptpjcn.1 𝑌 = 𝐽
ptpjcn.2 𝐽 = (∏t𝐹)
Assertion
Ref Expression
ptpjcn ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) → (𝑥𝑌 ↦ (𝑥𝐼)) ∈ (𝐽 Cn (𝐹𝐼)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐼   𝑥,𝑉   𝑥,𝑌
Allowed substitution hint:   𝐽(𝑥)

Proof of Theorem ptpjcn
Dummy variables 𝑔 𝑘 𝑢 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptpjcn.2 . . . . . 6 𝐽 = (∏t𝐹)
21ptuni 22130 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) = 𝐽)
323adant3 1124 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) → X𝑘𝐴 (𝐹𝑘) = 𝐽)
4 ptpjcn.1 . . . 4 𝑌 = 𝐽
53, 4syl6reqr 2872 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) → 𝑌 = X𝑘𝐴 (𝐹𝑘))
65mpteq1d 5146 . 2 ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) → (𝑥𝑌 ↦ (𝑥𝐼)) = (𝑥X𝑘𝐴 (𝐹𝑘) ↦ (𝑥𝐼)))
7 pttop 22118 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → (∏t𝐹) ∈ Top)
873adant3 1124 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) → (∏t𝐹) ∈ Top)
91, 8eqeltrid 2914 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) → 𝐽 ∈ Top)
10 ffvelrn 6841 . . . 4 ((𝐹:𝐴⟶Top ∧ 𝐼𝐴) → (𝐹𝐼) ∈ Top)
11103adant1 1122 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) → (𝐹𝐼) ∈ Top)
12 vex 3495 . . . . . . . . . 10 𝑥 ∈ V
1312elixp 8456 . . . . . . . . 9 (𝑥X𝑘𝐴 (𝐹𝑘) ↔ (𝑥 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑥𝑘) ∈ (𝐹𝑘)))
1413simprbi 497 . . . . . . . 8 (𝑥X𝑘𝐴 (𝐹𝑘) → ∀𝑘𝐴 (𝑥𝑘) ∈ (𝐹𝑘))
15 fveq2 6663 . . . . . . . . . 10 (𝑘 = 𝐼 → (𝑥𝑘) = (𝑥𝐼))
16 fveq2 6663 . . . . . . . . . . 11 (𝑘 = 𝐼 → (𝐹𝑘) = (𝐹𝐼))
1716unieqd 4840 . . . . . . . . . 10 (𝑘 = 𝐼 (𝐹𝑘) = (𝐹𝐼))
1815, 17eleq12d 2904 . . . . . . . . 9 (𝑘 = 𝐼 → ((𝑥𝑘) ∈ (𝐹𝑘) ↔ (𝑥𝐼) ∈ (𝐹𝐼)))
1918rspcva 3618 . . . . . . . 8 ((𝐼𝐴 ∧ ∀𝑘𝐴 (𝑥𝑘) ∈ (𝐹𝑘)) → (𝑥𝐼) ∈ (𝐹𝐼))
2014, 19sylan2 592 . . . . . . 7 ((𝐼𝐴𝑥X𝑘𝐴 (𝐹𝑘)) → (𝑥𝐼) ∈ (𝐹𝐼))
21203ad2antl3 1179 . . . . . 6 (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑥X𝑘𝐴 (𝐹𝑘)) → (𝑥𝐼) ∈ (𝐹𝐼))
2221fmpttd 6871 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) → (𝑥X𝑘𝐴 (𝐹𝑘) ↦ (𝑥𝐼)):X𝑘𝐴 (𝐹𝑘)⟶ (𝐹𝐼))
235feq2d 6493 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) → ((𝑥X𝑘𝐴 (𝐹𝑘) ↦ (𝑥𝐼)):𝑌 (𝐹𝐼) ↔ (𝑥X𝑘𝐴 (𝐹𝑘) ↦ (𝑥𝐼)):X𝑘𝐴 (𝐹𝑘)⟶ (𝐹𝐼)))
2422, 23mpbird 258 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) → (𝑥X𝑘𝐴 (𝐹𝑘) ↦ (𝑥𝐼)):𝑌 (𝐹𝐼))
25 eqid 2818 . . . . . . . . . . . 12 {𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑤 = X𝑦𝐴 (𝑔𝑦))} = {𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑤 = X𝑦𝐴 (𝑔𝑦))}
2625ptbas 22115 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝐴⟶Top) → {𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑤 = X𝑦𝐴 (𝑔𝑦))} ∈ TopBases)
27 bastg 21502 . . . . . . . . . . 11 ({𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑤 = X𝑦𝐴 (𝑔𝑦))} ∈ TopBases → {𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑤 = X𝑦𝐴 (𝑔𝑦))} ⊆ (topGen‘{𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑤 = X𝑦𝐴 (𝑔𝑦))}))
2826, 27syl 17 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝐴⟶Top) → {𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑤 = X𝑦𝐴 (𝑔𝑦))} ⊆ (topGen‘{𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑤 = X𝑦𝐴 (𝑔𝑦))}))
29 ffn 6507 . . . . . . . . . . 11 (𝐹:𝐴⟶Top → 𝐹 Fn 𝐴)
3025ptval 22106 . . . . . . . . . . . 12 ((𝐴𝑉𝐹 Fn 𝐴) → (∏t𝐹) = (topGen‘{𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑤 = X𝑦𝐴 (𝑔𝑦))}))
311, 30syl5eq 2865 . . . . . . . . . . 11 ((𝐴𝑉𝐹 Fn 𝐴) → 𝐽 = (topGen‘{𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑤 = X𝑦𝐴 (𝑔𝑦))}))
3229, 31sylan2 592 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐽 = (topGen‘{𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑤 = X𝑦𝐴 (𝑔𝑦))}))
3328, 32sseqtrrd 4005 . . . . . . . . 9 ((𝐴𝑉𝐹:𝐴⟶Top) → {𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑤 = X𝑦𝐴 (𝑔𝑦))} ⊆ 𝐽)
3433adantr 481 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝐼𝐴𝑢 ∈ (𝐹𝐼))) → {𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑤 = X𝑦𝐴 (𝑔𝑦))} ⊆ 𝐽)
35 eqid 2818 . . . . . . . . 9 X𝑘𝐴 (𝐹𝑘) = X𝑘𝐴 (𝐹𝑘)
3625, 35ptpjpre2 22116 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝐼𝐴𝑢 ∈ (𝐹𝐼))) → ((𝑥X𝑘𝐴 (𝐹𝑘) ↦ (𝑥𝐼)) “ 𝑢) ∈ {𝑤 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑤 = X𝑦𝐴 (𝑔𝑦))})
3734, 36sseldd 3965 . . . . . . 7 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝐼𝐴𝑢 ∈ (𝐹𝐼))) → ((𝑥X𝑘𝐴 (𝐹𝑘) ↦ (𝑥𝐼)) “ 𝑢) ∈ 𝐽)
3837expr 457 . . . . . 6 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ 𝐼𝐴) → (𝑢 ∈ (𝐹𝐼) → ((𝑥X𝑘𝐴 (𝐹𝑘) ↦ (𝑥𝐼)) “ 𝑢) ∈ 𝐽))
3938ralrimiv 3178 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ 𝐼𝐴) → ∀𝑢 ∈ (𝐹𝐼)((𝑥X𝑘𝐴 (𝐹𝑘) ↦ (𝑥𝐼)) “ 𝑢) ∈ 𝐽)
40393impa 1102 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) → ∀𝑢 ∈ (𝐹𝐼)((𝑥X𝑘𝐴 (𝐹𝑘) ↦ (𝑥𝐼)) “ 𝑢) ∈ 𝐽)
4124, 40jca 512 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) → ((𝑥X𝑘𝐴 (𝐹𝑘) ↦ (𝑥𝐼)):𝑌 (𝐹𝐼) ∧ ∀𝑢 ∈ (𝐹𝐼)((𝑥X𝑘𝐴 (𝐹𝑘) ↦ (𝑥𝐼)) “ 𝑢) ∈ 𝐽))
42 eqid 2818 . . . 4 (𝐹𝐼) = (𝐹𝐼)
434, 42iscn2 21774 . . 3 ((𝑥X𝑘𝐴 (𝐹𝑘) ↦ (𝑥𝐼)) ∈ (𝐽 Cn (𝐹𝐼)) ↔ ((𝐽 ∈ Top ∧ (𝐹𝐼) ∈ Top) ∧ ((𝑥X𝑘𝐴 (𝐹𝑘) ↦ (𝑥𝐼)):𝑌 (𝐹𝐼) ∧ ∀𝑢 ∈ (𝐹𝐼)((𝑥X𝑘𝐴 (𝐹𝑘) ↦ (𝑥𝐼)) “ 𝑢) ∈ 𝐽)))
449, 11, 41, 43syl21anbrc 1336 . 2 ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) → (𝑥X𝑘𝐴 (𝐹𝑘) ↦ (𝑥𝐼)) ∈ (𝐽 Cn (𝐹𝐼)))
456, 44eqeltrd 2910 1 ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) → (𝑥𝑌 ↦ (𝑥𝐼)) ∈ (𝐽 Cn (𝐹𝐼)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wral 3135  wrex 3136  cdif 3930  wss 3933   cuni 4830  cmpt 5137  ccnv 5547  cima 5551   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  Xcixp 8449  Fincfn 8497  topGenctg 16699  tcpt 16700  Topctop 21429  TopBasesctb 21481   Cn ccn 21760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-ixp 8450  df-en 8498  df-fin 8501  df-fi 8863  df-topgen 16705  df-pt 16706  df-top 21430  df-topon 21447  df-bases 21482  df-cn 21763
This theorem is referenced by:  pthaus  22174  ptrescn  22175  xkopjcn  22192  pt1hmeo  22342  ptunhmeo  22344  tmdgsum  22631  symgtgp  22637  prdstmdd  22659  prdstgpd  22660  poimir  34806  broucube  34807
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