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Theorem ptval2 21903
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 6338 . . 3 (𝐹:𝐴⟶Top → 𝐹 Fn 𝐴)
2 ptval2.1 . . . 4 𝐽 = (∏t𝐹)
3 eqid 2772 . . . . 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 21872 . . . 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 2820 . . 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 583 . 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 2772 . . . . 5 X𝑛𝐴 (𝐹𝑛) = X𝑛𝐴 (𝐹𝑛)
83, 7ptbasfi 21883 . . . 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 21896 . . . . . . . 8 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = 𝐽)
10 ptval2.2 . . . . . . . 8 𝑋 = 𝐽
119, 10syl6eqr 2826 . . . . . . 7 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = 𝑋)
1211sneqd 4447 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Top) → {X𝑛𝐴 (𝐹𝑛)} = {𝑋})
13113ad2ant1 1113 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ 𝑘𝐴𝑢 ∈ (𝐹𝑘)) → X𝑛𝐴 (𝐹𝑛) = 𝑋)
1413mpteq1d 5010 . . . . . . . . . . 11 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ 𝑘𝐴𝑢 ∈ (𝐹𝑘)) → (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑘)))
1514cnveqd 5589 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ 𝑘𝐴𝑢 ∈ (𝐹𝑘)) → (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑘)))
1615imaeq1d 5763 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ 𝑘𝐴𝑢 ∈ (𝐹𝑘)) → ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
1716mpoeq3dva 7043 . . . . . . . 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, 18syl6eqr 2826 . . . . . . 7 ((𝐴𝑉𝐹:𝐴⟶Top) → (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)) = 𝐺)
2019rneqd 5644 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Top) → ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)) = ran 𝐺)
2112, 20uneq12d 4025 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → ({X𝑛𝐴 (𝐹𝑛)} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢))) = ({𝑋} ∪ ran 𝐺))
2221fveq2d 6497 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘({X𝑛𝐴 (𝐹𝑛)} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)))) = (fi‘({𝑋} ∪ ran 𝐺)))
238, 22eqtrd 2808 . . 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 6497 . 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 2808 1 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐽 = (topGen‘(fi‘({𝑋} ∪ ran 𝐺))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wex 1742  wcel 2048  {cab 2753  wral 3082  wrex 3083  cdif 3822  cun 3823  {csn 4435   cuni 4706  cmpt 5002  ccnv 5399  ran crn 5401  cima 5403   Fn wfn 6177  wf 6178  cfv 6182  cmpo 6972  Xcixp 8251  Fincfn 8298  ficfi 8661  topGenctg 16557  tcpt 16558  Topctop 21195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7494  df-2nd 7495  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-oadd 7901  df-er 8081  df-ixp 8252  df-en 8299  df-dom 8300  df-fin 8302  df-fi 8662  df-topgen 16563  df-pt 16564  df-top 21196  df-bases 21248
This theorem is referenced by:  ptrescn  21941  ptrest  34280
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