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Theorem ptcmplem1 22588
Description: Lemma for ptcmp 22594. (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 6508 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
4 eqid 2818 . . . . . . . 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 22106 . . . . . . 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 21931 . . . . . . . . . . 11 (𝑥 ∈ Comp → 𝑥 ∈ Top)
87ssriv 3968 . . . . . . . . . 10 Comp ⊆ Top
9 fss 6520 . . . . . . . . . 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 22117 . . . . . . . . 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 4126 . . . . . . . . . 10 (ran 𝑆 ∪ {𝑋}) = ({𝑋} ∪ ran 𝑆)
15 ptcmp.1 . . . . . . . . . . . 12 𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
1615rneqi 5800 . . . . . . . . . . 11 ran 𝑆 = ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
1716uneq2i 4133 . . . . . . . . . 10 ({𝑋} ∪ ran 𝑆) = ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
1814, 17eqtri 2841 . . . . . . . . 9 (ran 𝑆 ∪ {𝑋}) = ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
1918fveq2i 6666 . . . . . . . 8 (fi‘(ran 𝑆 ∪ {𝑋})) = (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
2013, 19syl6eqr 2871 . . . . . . 7 (𝜑 → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} = (fi‘(ran 𝑆 ∪ {𝑋})))
2120fveq2d 6667 . . . . . 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 2853 . . . . 5 (𝜑 → (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
2322unieqd 4840 . . . 4 (𝜑 (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
24 fibas 21513 . . . . 5 (fi‘(ran 𝑆 ∪ {𝑋})) ∈ TopBases
25 unitg 21503 . . . . 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, 26syl6eq 2869 . . 3 (𝜑 (∏t𝐹) = (fi‘(ran 𝑆 ∪ {𝑋})))
28 eqid 2818 . . . . . 6 (∏t𝐹) = (∏t𝐹)
2928ptuni 22130 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = (∏t𝐹))
301, 10, 29syl2anc 584 . . . 4 (𝜑X𝑛𝐴 (𝐹𝑛) = (∏t𝐹))
3111, 30syl5eq 2865 . . 3 (𝜑𝑋 = (∏t𝐹))
32 ptcmp.5 . . . . . . 7 (𝜑𝑋 ∈ (UFL ∩ dom card))
3332pwexd 5271 . . . . . 6 (𝜑 → 𝒫 𝑋 ∈ V)
34 eqid 2818 . . . . . . . . . . . 12 (𝑤𝑋 ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑘))
3534mptpreima 6085 . . . . . . . . . . 11 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) = {𝑤𝑋 ∣ (𝑤𝑘) ∈ 𝑢}
3635ssrab3 4054 . . . . . . . . . 10 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ⊆ 𝑋
3732adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘𝐴𝑢 ∈ (𝐹𝑘))) → 𝑋 ∈ (UFL ∩ dom card))
38 elpw2g 5238 . . . . . . . . . . 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 259 . . . . . . . . 9 ((𝜑 ∧ (𝑘𝐴𝑢 ∈ (𝐹𝑘))) → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋)
4140ralrimivva 3188 . . . . . . . 8 (𝜑 → ∀𝑘𝐴𝑢 ∈ (𝐹𝑘)((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋)
4215fmpox 7754 . . . . . . . 8 (∀𝑘𝐴𝑢 ∈ (𝐹𝑘)((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋𝑆: 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝒫 𝑋)
4341, 42sylib 219 . . . . . . 7 (𝜑𝑆: 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝒫 𝑋)
4443frnd 6514 . . . . . 6 (𝜑 → ran 𝑆 ⊆ 𝒫 𝑋)
4533, 44ssexd 5219 . . . . 5 (𝜑 → ran 𝑆 ∈ V)
46 snex 5322 . . . . 5 {𝑋} ∈ V
47 unexg 7461 . . . . 5 ((ran 𝑆 ∈ V ∧ {𝑋} ∈ V) → (ran 𝑆 ∪ {𝑋}) ∈ V)
4845, 46, 47sylancl 586 . . . 4 (𝜑 → (ran 𝑆 ∪ {𝑋}) ∈ V)
49 fiuni 8880 . . . 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 2863 . 2 (𝜑𝑋 = (ran 𝑆 ∪ {𝑋}))
5251, 22jca 512 1 (𝜑 → (𝑋 = (ran 𝑆 ∪ {𝑋}) ∧ (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋})))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wral 3135  wrex 3136  Vcvv 3492  cdif 3930  cun 3931  cin 3932  wss 3933  𝒫 cpw 4535  {csn 4557   cuni 4830   ciun 4910  cmpt 5137   × cxp 5546  ccnv 5547  dom cdm 5548  ran crn 5549  cima 5551   Fn wfn 6343  wf 6344  cfv 6348  cmpo 7147  Xcixp 8449  Fincfn 8497  ficfi 8862  cardccrd 9352  topGenctg 16699  tcpt 16700  Topctop 21429  TopBasesctb 21481  Compccmp 21922  UFLcufl 22436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-ixp 8450  df-en 8498  df-dom 8499  df-fin 8501  df-fi 8863  df-topgen 16705  df-pt 16706  df-top 21430  df-bases 21482  df-cmp 21923
This theorem is referenced by:  ptcmplem5  22592
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