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Theorem ptval2 22498
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 6545 . . 3 (𝐹:𝐴⟶Top → 𝐹 Fn 𝐴)
2 ptval2.1 . . . 4 𝐽 = (∏t𝐹)
3 eqid 2737 . . . . 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 22467 . . . 4 ((𝐴𝑉𝐹 Fn 𝐴) → (∏t𝐹) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
52, 4syl5eq 2790 . . 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 596 . 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 2737 . . . . 5 X𝑛𝐴 (𝐹𝑛) = X𝑛𝐴 (𝐹𝑛)
83, 7ptbasfi 22478 . . . 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 22491 . . . . . . . 8 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = 𝐽)
10 ptval2.2 . . . . . . . 8 𝑋 = 𝐽
119, 10eqtr4di 2796 . . . . . . 7 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = 𝑋)
1211sneqd 4553 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Top) → {X𝑛𝐴 (𝐹𝑛)} = {𝑋})
13113ad2ant1 1135 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ 𝑘𝐴𝑢 ∈ (𝐹𝑘)) → X𝑛𝐴 (𝐹𝑛) = 𝑋)
1413mpteq1d 5144 . . . . . . . . . . 11 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ 𝑘𝐴𝑢 ∈ (𝐹𝑘)) → (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑘)))
1514cnveqd 5744 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ 𝑘𝐴𝑢 ∈ (𝐹𝑘)) → (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑘)))
1615imaeq1d 5928 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ 𝑘𝐴𝑢 ∈ (𝐹𝑘)) → ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
1716mpoeq3dva 7288 . . . . . . . 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 2796 . . . . . . 7 ((𝐴𝑉𝐹:𝐴⟶Top) → (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)) = 𝐺)
2019rneqd 5807 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Top) → ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)) = ran 𝐺)
2112, 20uneq12d 4078 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → ({X𝑛𝐴 (𝐹𝑛)} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢))) = ({𝑋} ∪ ran 𝐺))
2221fveq2d 6721 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘({X𝑛𝐴 (𝐹𝑛)} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)))) = (fi‘({𝑋} ∪ ran 𝐺)))
238, 22eqtrd 2777 . . 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 6721 . 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 2777 1 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐽 = (topGen‘(fi‘({𝑋} ∪ ran 𝐺))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wex 1787  wcel 2110  {cab 2714  wral 3061  wrex 3062  cdif 3863  cun 3864  {csn 4541   cuni 4819  cmpt 5135  ccnv 5550  ran crn 5552  cima 5554   Fn wfn 6375  wf 6376  cfv 6380  cmpo 7215  Xcixp 8578  Fincfn 8626  ficfi 9026  topGenctg 16942  tcpt 16943  Topctop 21790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-1o 8202  df-er 8391  df-ixp 8579  df-en 8627  df-dom 8628  df-fin 8630  df-fi 9027  df-topgen 16948  df-pt 16949  df-top 21791  df-bases 21843
This theorem is referenced by:  ptrescn  22536  ptrest  35513
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