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Theorem ptcmplem1 24076
Description: Lemma for ptcmp 24082. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypotheses
Ref Expression
ptcmp.1 𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
ptcmp.2 𝑋 = X𝑛𝐴 (𝐹𝑛)
ptcmp.3 (𝜑𝐴𝑉)
ptcmp.4 (𝜑𝐹:𝐴⟶Comp)
ptcmp.5 (𝜑𝑋 ∈ (UFL ∩ dom card))
Assertion
Ref Expression
ptcmplem1 (𝜑 → (𝑋 = (ran 𝑆 ∪ {𝑋}) ∧ (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋})))))
Distinct variable groups:   𝑘,𝑛,𝑢,𝑤,𝐴   𝑆,𝑘,𝑛,𝑢   𝜑,𝑘,𝑛,𝑢   𝑘,𝑉,𝑛,𝑢,𝑤   𝑘,𝐹,𝑛,𝑢,𝑤   𝑘,𝑋,𝑛,𝑢,𝑤
Allowed substitution hints:   𝜑(𝑤)   𝑆(𝑤)

Proof of Theorem ptcmplem1
Dummy variables 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptcmp.3 . . . . . . 7 (𝜑𝐴𝑉)
2 ptcmp.4 . . . . . . . 8 (𝜑𝐹:𝐴⟶Comp)
32ffnd 6738 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
4 eqid 2735 . . . . . . . 8 {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))}
54ptval 23594 . . . . . . 7 ((𝐴𝑉𝐹 Fn 𝐴) → (∏t𝐹) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
61, 3, 5syl2anc 584 . . . . . 6 (𝜑 → (∏t𝐹) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
7 cmptop 23419 . . . . . . . . . . 11 (𝑥 ∈ Comp → 𝑥 ∈ Top)
87ssriv 3999 . . . . . . . . . 10 Comp ⊆ Top
9 fss 6753 . . . . . . . . . 10 ((𝐹:𝐴⟶Comp ∧ Comp ⊆ Top) → 𝐹:𝐴⟶Top)
102, 8, 9sylancl 586 . . . . . . . . 9 (𝜑𝐹:𝐴⟶Top)
11 ptcmp.2 . . . . . . . . . 10 𝑋 = X𝑛𝐴 (𝐹𝑛)
124, 11ptbasfi 23605 . . . . . . . . 9 ((𝐴𝑉𝐹:𝐴⟶Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} = (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
131, 10, 12syl2anc 584 . . . . . . . 8 (𝜑 → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} = (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))))
14 uncom 4168 . . . . . . . . . 10 (ran 𝑆 ∪ {𝑋}) = ({𝑋} ∪ ran 𝑆)
15 ptcmp.1 . . . . . . . . . . . 12 𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
1615rneqi 5951 . . . . . . . . . . 11 ran 𝑆 = ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
1716uneq2i 4175 . . . . . . . . . 10 ({𝑋} ∪ ran 𝑆) = ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
1814, 17eqtri 2763 . . . . . . . . 9 (ran 𝑆 ∪ {𝑋}) = ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
1918fveq2i 6910 . . . . . . . 8 (fi‘(ran 𝑆 ∪ {𝑋})) = (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
2013, 19eqtr4di 2793 . . . . . . 7 (𝜑 → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} = (fi‘(ran 𝑆 ∪ {𝑋})))
2120fveq2d 6911 . . . . . 6 (𝜑 → (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
226, 21eqtrd 2775 . . . . 5 (𝜑 → (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
2322unieqd 4925 . . . 4 (𝜑 (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
24 fibas 23000 . . . . 5 (fi‘(ran 𝑆 ∪ {𝑋})) ∈ TopBases
25 unitg 22990 . . . . 5 ((fi‘(ran 𝑆 ∪ {𝑋})) ∈ TopBases → (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))) = (fi‘(ran 𝑆 ∪ {𝑋})))
2624, 25ax-mp 5 . . . 4 (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))) = (fi‘(ran 𝑆 ∪ {𝑋}))
2723, 26eqtrdi 2791 . . 3 (𝜑 (∏t𝐹) = (fi‘(ran 𝑆 ∪ {𝑋})))
28 eqid 2735 . . . . . 6 (∏t𝐹) = (∏t𝐹)
2928ptuni 23618 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = (∏t𝐹))
301, 10, 29syl2anc 584 . . . 4 (𝜑X𝑛𝐴 (𝐹𝑛) = (∏t𝐹))
3111, 30eqtrid 2787 . . 3 (𝜑𝑋 = (∏t𝐹))
32 ptcmp.5 . . . . . . 7 (𝜑𝑋 ∈ (UFL ∩ dom card))
3332pwexd 5385 . . . . . 6 (𝜑 → 𝒫 𝑋 ∈ V)
34 eqid 2735 . . . . . . . . . . . 12 (𝑤𝑋 ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑘))
3534mptpreima 6260 . . . . . . . . . . 11 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) = {𝑤𝑋 ∣ (𝑤𝑘) ∈ 𝑢}
3635ssrab3 4092 . . . . . . . . . 10 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ⊆ 𝑋
3732adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘𝐴𝑢 ∈ (𝐹𝑘))) → 𝑋 ∈ (UFL ∩ dom card))
38 elpw2g 5339 . . . . . . . . . . 11 (𝑋 ∈ (UFL ∩ dom card) → (((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋 ↔ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ⊆ 𝑋))
3937, 38syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑘𝐴𝑢 ∈ (𝐹𝑘))) → (((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋 ↔ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ⊆ 𝑋))
4036, 39mpbiri 258 . . . . . . . . 9 ((𝜑 ∧ (𝑘𝐴𝑢 ∈ (𝐹𝑘))) → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋)
4140ralrimivva 3200 . . . . . . . 8 (𝜑 → ∀𝑘𝐴𝑢 ∈ (𝐹𝑘)((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋)
4215fmpox 8091 . . . . . . . 8 (∀𝑘𝐴𝑢 ∈ (𝐹𝑘)((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋𝑆: 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝒫 𝑋)
4341, 42sylib 218 . . . . . . 7 (𝜑𝑆: 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝒫 𝑋)
4443frnd 6745 . . . . . 6 (𝜑 → ran 𝑆 ⊆ 𝒫 𝑋)
4533, 44ssexd 5330 . . . . 5 (𝜑 → ran 𝑆 ∈ V)
46 snex 5442 . . . . 5 {𝑋} ∈ V
47 unexg 7762 . . . . 5 ((ran 𝑆 ∈ V ∧ {𝑋} ∈ V) → (ran 𝑆 ∪ {𝑋}) ∈ V)
4845, 46, 47sylancl 586 . . . 4 (𝜑 → (ran 𝑆 ∪ {𝑋}) ∈ V)
49 fiuni 9466 . . . 4 ((ran 𝑆 ∪ {𝑋}) ∈ V → (ran 𝑆 ∪ {𝑋}) = (fi‘(ran 𝑆 ∪ {𝑋})))
5048, 49syl 17 . . 3 (𝜑 (ran 𝑆 ∪ {𝑋}) = (fi‘(ran 𝑆 ∪ {𝑋})))
5127, 31, 503eqtr4d 2785 . 2 (𝜑𝑋 = (ran 𝑆 ∪ {𝑋}))
5251, 22jca 511 1 (𝜑 → (𝑋 = (ran 𝑆 ∪ {𝑋}) ∧ (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋})))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  cdif 3960  cun 3961  cin 3962  wss 3963  𝒫 cpw 4605  {csn 4631   cuni 4912   ciun 4996  cmpt 5231   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692   Fn wfn 6558  wf 6559  cfv 6563  cmpo 7433  Xcixp 8936  Fincfn 8984  ficfi 9448  cardccrd 9973  topGenctg 17484  tcpt 17485  Topctop 22915  TopBasesctb 22968  Compccmp 23410  UFLcufl 23924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-ixp 8937  df-en 8985  df-dom 8986  df-fin 8988  df-fi 9449  df-topgen 17490  df-pt 17491  df-top 22916  df-bases 22969  df-cmp 23411
This theorem is referenced by:  ptcmplem5  24080
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