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Theorem ptval2 21317
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 6004 . . 3 (𝐹:𝐴⟶Top → 𝐹 Fn 𝐴)
2 ptval2.1 . . . 4 𝐽 = (∏t𝐹)
3 eqid 2621 . . . . 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 21286 . . . 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 2667 . . 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 491 . 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 2621 . . . . 5 X𝑛𝐴 (𝐹𝑛) = X𝑛𝐴 (𝐹𝑛)
83, 7ptbasfi 21297 . . . 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 21310 . . . . . . . 8 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = 𝐽)
10 ptval2.2 . . . . . . . 8 𝑋 = 𝐽
119, 10syl6eqr 2673 . . . . . . 7 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = 𝑋)
1211sneqd 4162 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Top) → {X𝑛𝐴 (𝐹𝑛)} = {𝑋})
13113ad2ant1 1080 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ 𝑘𝐴𝑢 ∈ (𝐹𝑘)) → X𝑛𝐴 (𝐹𝑛) = 𝑋)
1413mpteq1d 4700 . . . . . . . . . . 11 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ 𝑘𝐴𝑢 ∈ (𝐹𝑘)) → (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑘)))
1514cnveqd 5260 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ 𝑘𝐴𝑢 ∈ (𝐹𝑘)) → (𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑘)))
1615imaeq1d 5426 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ 𝑘𝐴𝑢 ∈ (𝐹𝑘)) → ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
1716mpt2eq3dva 6675 . . . . . . . 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 2673 . . . . . . 7 ((𝐴𝑉𝐹:𝐴⟶Top) → (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)) = 𝐺)
2019rneqd 5315 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Top) → ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)) = ran 𝐺)
2112, 20uneq12d 3748 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → ({X𝑛𝐴 (𝐹𝑛)} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢))) = ({𝑋} ∪ ran 𝐺))
2221fveq2d 6154 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Top) → (fi‘({X𝑛𝐴 (𝐹𝑛)} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤X𝑛𝐴 (𝐹𝑛) ↦ (𝑤𝑘)) “ 𝑢)))) = (fi‘({𝑋} ∪ ran 𝐺)))
238, 22eqtrd 2655 . . 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 6154 . 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 2655 1 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐽 = (topGen‘(fi‘({𝑋} ∪ ran 𝐺))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  wrex 2908  cdif 3553  cun 3554  {csn 4150   cuni 4404  cmpt 4675  ccnv 5075  ran crn 5077  cima 5079   Fn wfn 5844  wf 5845  cfv 5849  cmpt2 6609  Xcixp 7855  Fincfn 7902  ficfi 8263  topGenctg 16022  tcpt 16023  Topctop 20620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-iin 4490  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-oadd 7512  df-er 7690  df-ixp 7856  df-en 7903  df-dom 7904  df-fin 7906  df-fi 8264  df-topgen 16028  df-pt 16029  df-top 20621  df-bases 20664
This theorem is referenced by:  ptrescn  21355  ptrest  33061
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