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Theorem ptcmplem1 21903
Description: Lemma for ptcmp 21909. (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)
3 ffn 6083 . . . . . . . 8 (𝐹:𝐴⟶Comp → 𝐹 Fn 𝐴)
42, 3syl 17 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
5 eqid 2651 . . . . . . . 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))}
65ptval 21421 . . . . . . 7 ((𝐴𝑉𝐹 Fn 𝐴) → (∏t𝐹) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
71, 4, 6syl2anc 694 . . . . . 6 (𝜑 → (∏t𝐹) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
8 cmptop 21246 . . . . . . . . . . 11 (𝑥 ∈ Comp → 𝑥 ∈ Top)
98ssriv 3640 . . . . . . . . . 10 Comp ⊆ Top
10 fss 6094 . . . . . . . . . 10 ((𝐹:𝐴⟶Comp ∧ Comp ⊆ Top) → 𝐹:𝐴⟶Top)
112, 9, 10sylancl 695 . . . . . . . . 9 (𝜑𝐹:𝐴⟶Top)
12 ptcmp.2 . . . . . . . . . 10 𝑋 = X𝑛𝐴 (𝐹𝑛)
135, 12ptbasfi 21432 . . . . . . . . 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)))))
141, 11, 13syl2anc 694 . . . . . . . 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)))))
15 uncom 3790 . . . . . . . . . 10 (ran 𝑆 ∪ {𝑋}) = ({𝑋} ∪ ran 𝑆)
16 ptcmp.1 . . . . . . . . . . . 12 𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
1716rneqi 5384 . . . . . . . . . . 11 ran 𝑆 = ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
1817uneq2i 3797 . . . . . . . . . 10 ({𝑋} ∪ ran 𝑆) = ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
1915, 18eqtri 2673 . . . . . . . . 9 (ran 𝑆 ∪ {𝑋}) = ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
2019fveq2i 6232 . . . . . . . 8 (fi‘(ran 𝑆 ∪ {𝑋})) = (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
2114, 20syl6eqr 2703 . . . . . . 7 (𝜑 → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} = (fi‘(ran 𝑆 ∪ {𝑋})))
2221fveq2d 6233 . . . . . 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}))))
237, 22eqtrd 2685 . . . . 5 (𝜑 → (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
2423unieqd 4478 . . . 4 (𝜑 (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
25 fibas 20829 . . . . 5 (fi‘(ran 𝑆 ∪ {𝑋})) ∈ TopBases
26 unitg 20819 . . . . 5 ((fi‘(ran 𝑆 ∪ {𝑋})) ∈ TopBases → (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))) = (fi‘(ran 𝑆 ∪ {𝑋})))
2725, 26ax-mp 5 . . . 4 (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))) = (fi‘(ran 𝑆 ∪ {𝑋}))
2824, 27syl6eq 2701 . . 3 (𝜑 (∏t𝐹) = (fi‘(ran 𝑆 ∪ {𝑋})))
29 eqid 2651 . . . . . 6 (∏t𝐹) = (∏t𝐹)
3029ptuni 21445 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = (∏t𝐹))
311, 11, 30syl2anc 694 . . . 4 (𝜑X𝑛𝐴 (𝐹𝑛) = (∏t𝐹))
3212, 31syl5eq 2697 . . 3 (𝜑𝑋 = (∏t𝐹))
33 ptcmp.5 . . . . . . 7 (𝜑𝑋 ∈ (UFL ∩ dom card))
34 pwexg 4880 . . . . . . 7 (𝑋 ∈ (UFL ∩ dom card) → 𝒫 𝑋 ∈ V)
3533, 34syl 17 . . . . . 6 (𝜑 → 𝒫 𝑋 ∈ V)
36 eqid 2651 . . . . . . . . . . . 12 (𝑤𝑋 ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑘))
3736mptpreima 5666 . . . . . . . . . . 11 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) = {𝑤𝑋 ∣ (𝑤𝑘) ∈ 𝑢}
38 ssrab2 3720 . . . . . . . . . . 11 {𝑤𝑋 ∣ (𝑤𝑘) ∈ 𝑢} ⊆ 𝑋
3937, 38eqsstri 3668 . . . . . . . . . 10 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ⊆ 𝑋
4033adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘𝐴𝑢 ∈ (𝐹𝑘))) → 𝑋 ∈ (UFL ∩ dom card))
41 elpw2g 4857 . . . . . . . . . . 11 (𝑋 ∈ (UFL ∩ dom card) → (((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋 ↔ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ⊆ 𝑋))
4240, 41syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑘𝐴𝑢 ∈ (𝐹𝑘))) → (((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋 ↔ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ⊆ 𝑋))
4339, 42mpbiri 248 . . . . . . . . 9 ((𝜑 ∧ (𝑘𝐴𝑢 ∈ (𝐹𝑘))) → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋)
4443ralrimivva 3000 . . . . . . . 8 (𝜑 → ∀𝑘𝐴𝑢 ∈ (𝐹𝑘)((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋)
4516fmpt2x 7281 . . . . . . . 8 (∀𝑘𝐴𝑢 ∈ (𝐹𝑘)((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋𝑆: 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝒫 𝑋)
4644, 45sylib 208 . . . . . . 7 (𝜑𝑆: 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝒫 𝑋)
47 frn 6091 . . . . . . 7 (𝑆: 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝒫 𝑋 → ran 𝑆 ⊆ 𝒫 𝑋)
4846, 47syl 17 . . . . . 6 (𝜑 → ran 𝑆 ⊆ 𝒫 𝑋)
4935, 48ssexd 4838 . . . . 5 (𝜑 → ran 𝑆 ∈ V)
50 snex 4938 . . . . 5 {𝑋} ∈ V
51 unexg 7001 . . . . 5 ((ran 𝑆 ∈ V ∧ {𝑋} ∈ V) → (ran 𝑆 ∪ {𝑋}) ∈ V)
5249, 50, 51sylancl 695 . . . 4 (𝜑 → (ran 𝑆 ∪ {𝑋}) ∈ V)
53 fiuni 8375 . . . 4 ((ran 𝑆 ∪ {𝑋}) ∈ V → (ran 𝑆 ∪ {𝑋}) = (fi‘(ran 𝑆 ∪ {𝑋})))
5452, 53syl 17 . . 3 (𝜑 (ran 𝑆 ∪ {𝑋}) = (fi‘(ran 𝑆 ∪ {𝑋})))
5528, 32, 543eqtr4d 2695 . 2 (𝜑𝑋 = (ran 𝑆 ∪ {𝑋}))
5655, 23jca 553 1 (𝜑 → (𝑋 = (ran 𝑆 ∪ {𝑋}) ∧ (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋})))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  𝒫 cpw 4191  {csn 4210   cuni 4468   ciun 4552  cmpt 4762   × cxp 5141  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146   Fn wfn 5921  wf 5922  cfv 5926  cmpt2 6692  Xcixp 7950  Fincfn 7997  ficfi 8357  cardccrd 8799  topGenctg 16145  tcpt 16146  Topctop 20746  TopBasesctb 20797  Compccmp 21237  UFLcufl 21751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-ixp 7951  df-en 7998  df-dom 7999  df-fin 8001  df-fi 8358  df-topgen 16151  df-pt 16152  df-top 20747  df-bases 20798  df-cmp 21238
This theorem is referenced by:  ptcmplem5  21907
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