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Theorem ptval2 23630
Description: The value of the product topology function. (Contributed by Mario Carneiro, 7-Feb-2015.)
Hypotheses
Ref Expression
ptval2.1 𝐽 = (∏t𝐹)
ptval2.2 𝑋 = 𝐽
ptval2.3 𝐺 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
Assertion
Ref Expression
ptval2 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐽 = (topGen‘(fi‘({𝑋} ∪ ran 𝐺))))
Distinct variable groups:   𝑢,𝑘,𝑤,𝐴   𝑘,𝐹,𝑢,𝑤   𝑘,𝑉,𝑢,𝑤   𝑤,𝑋
Allowed substitution hints:   𝐺(𝑤,𝑢,𝑘)   𝐽(𝑤,𝑢,𝑘)   𝑋(𝑢,𝑘)

Proof of Theorem ptval2
Dummy variables 𝑔 𝑛 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffn 6747 . . 3 (𝐹:𝐴⟶Top → 𝐹 Fn 𝐴)
2 ptval2.1 . . . 4 𝐽 = (∏t𝐹)
3 eqid 2740 . . . . 5 {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
43ptval 23599 . . . 4 ((𝐴𝑉𝐹 Fn 𝐴) → (∏t𝐹) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
52, 4eqtrid 2792 . . 3 ((𝐴𝑉𝐹 Fn 𝐴) → 𝐽 = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
61, 5sylan2 592 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐽 = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
7 eqid 2740 . . . . 5 X𝑛𝐴 (𝐹𝑛) = X𝑛𝐴 (𝐹𝑛)
83, 7ptbasfi 23610 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} = (fi‘({X𝑛𝐴 (𝐹𝑛)} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)))))
92ptuni 23623 . . . . . . . 8 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = 𝐽)
10 ptval2.2 . . . . . . . 8 𝑋 = 𝐽
119, 10eqtr4di 2798 . . . . . . 7 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = 𝑋)
1211sneqd 4660 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Top) → {X𝑛𝐴 (𝐹𝑛)} = {𝑋})
13113ad2ant1 1133 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ 𝑘𝐴𝑢 ∈ (𝐹𝑘)) → X𝑛𝐴 (𝐹𝑛) = 𝑋)
1413mpteq1d 5261 . . . . . . . . . . 11 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ 𝑘𝐴𝑢 ∈ (𝐹𝑘)) → (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑘)))
1514cnveqd 5900 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ 𝑘𝐴𝑢 ∈ (𝐹𝑘)) → (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑘)))
1615imaeq1d 6088 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ 𝑘𝐴𝑢 ∈ (𝐹𝑘)) → ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
1716mpoeq3dva 7527 . . . . . . . 8 ((𝐴𝑉𝐹:𝐴⟶Top) → (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)) = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
18 ptval2.3 . . . . . . . 8 𝐺 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
1917, 18eqtr4di 2798 . . . . . . 7 ((𝐴𝑉𝐹:𝐴⟶Top) → (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)) = 𝐺)
2019rneqd 5963 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Top) → ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)) = ran 𝐺)
2112, 20uneq12d 4192 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → ({X𝑛𝐴 (𝐹𝑛)} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢))) = ({𝑋} ∪ ran 𝐺))
2221fveq2d 6924 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘({X𝑛𝐴 (𝐹𝑛)} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)))) = (fi‘({𝑋} ∪ ran 𝐺)))
238, 22eqtrd 2780 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} = (fi‘({𝑋} ∪ ran 𝐺)))
2423fveq2d 6924 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}) = (topGen‘(fi‘({𝑋} ∪ ran 𝐺))))
256, 24eqtrd 2780 1 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐽 = (topGen‘(fi‘({𝑋} ∪ ran 𝐺))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1537  wex 1777  wcel 2108  {cab 2717  wral 3067  wrex 3076  cdif 3973  cun 3974  {csn 4648   cuni 4931  cmpt 5249  ccnv 5699  ran crn 5701  cima 5703   Fn wfn 6568  wf 6569  cfv 6573  cmpo 7450  Xcixp 8955  Fincfn 9003  ficfi 9479  topGenctg 17497  tcpt 17498  Topctop 22920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-1o 8522  df-2o 8523  df-ixp 8956  df-en 9004  df-dom 9005  df-fin 9007  df-fi 9480  df-topgen 17503  df-pt 17504  df-top 22921  df-bases 22974
This theorem is referenced by:  ptrescn  23668  ptrest  37579
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