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Theorem ptcmplem1 24060
Description: Lemma for ptcmp 24066. (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 6737 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
4 eqid 2737 . . . . . . . 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 23578 . . . . . . 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 23403 . . . . . . . . . . 11 (𝑥 ∈ Comp → 𝑥 ∈ Top)
87ssriv 3987 . . . . . . . . . 10 Comp ⊆ Top
9 fss 6752 . . . . . . . . . 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 23589 . . . . . . . . 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 4158 . . . . . . . . . 10 (ran 𝑆 ∪ {𝑋}) = ({𝑋} ∪ ran 𝑆)
15 ptcmp.1 . . . . . . . . . . . 12 𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
1615rneqi 5948 . . . . . . . . . . 11 ran 𝑆 = ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
1716uneq2i 4165 . . . . . . . . . 10 ({𝑋} ∪ ran 𝑆) = ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
1814, 17eqtri 2765 . . . . . . . . 9 (ran 𝑆 ∪ {𝑋}) = ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
1918fveq2i 6909 . . . . . . . 8 (fi‘(ran 𝑆 ∪ {𝑋})) = (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
2013, 19eqtr4di 2795 . . . . . . 7 (𝜑 → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} = (fi‘(ran 𝑆 ∪ {𝑋})))
2120fveq2d 6910 . . . . . 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 2777 . . . . 5 (𝜑 → (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
2322unieqd 4920 . . . 4 (𝜑 (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
24 fibas 22984 . . . . 5 (fi‘(ran 𝑆 ∪ {𝑋})) ∈ TopBases
25 unitg 22974 . . . . 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 2793 . . 3 (𝜑 (∏t𝐹) = (fi‘(ran 𝑆 ∪ {𝑋})))
28 eqid 2737 . . . . . 6 (∏t𝐹) = (∏t𝐹)
2928ptuni 23602 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = (∏t𝐹))
301, 10, 29syl2anc 584 . . . 4 (𝜑X𝑛𝐴 (𝐹𝑛) = (∏t𝐹))
3111, 30eqtrid 2789 . . 3 (𝜑𝑋 = (∏t𝐹))
32 ptcmp.5 . . . . . . 7 (𝜑𝑋 ∈ (UFL ∩ dom card))
3332pwexd 5379 . . . . . 6 (𝜑 → 𝒫 𝑋 ∈ V)
34 eqid 2737 . . . . . . . . . . . 12 (𝑤𝑋 ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑘))
3534mptpreima 6258 . . . . . . . . . . 11 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) = {𝑤𝑋 ∣ (𝑤𝑘) ∈ 𝑢}
3635ssrab3 4082 . . . . . . . . . 10 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ⊆ 𝑋
3732adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘𝐴𝑢 ∈ (𝐹𝑘))) → 𝑋 ∈ (UFL ∩ dom card))
38 elpw2g 5333 . . . . . . . . . . 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 3202 . . . . . . . 8 (𝜑 → ∀𝑘𝐴𝑢 ∈ (𝐹𝑘)((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋)
4215fmpox 8092 . . . . . . . 8 (∀𝑘𝐴𝑢 ∈ (𝐹𝑘)((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋𝑆: 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝒫 𝑋)
4341, 42sylib 218 . . . . . . 7 (𝜑𝑆: 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝒫 𝑋)
4443frnd 6744 . . . . . 6 (𝜑 → ran 𝑆 ⊆ 𝒫 𝑋)
4533, 44ssexd 5324 . . . . 5 (𝜑 → ran 𝑆 ∈ V)
46 snex 5436 . . . . 5 {𝑋} ∈ V
47 unexg 7763 . . . . 5 ((ran 𝑆 ∈ V ∧ {𝑋} ∈ V) → (ran 𝑆 ∪ {𝑋}) ∈ V)
4845, 46, 47sylancl 586 . . . 4 (𝜑 → (ran 𝑆 ∪ {𝑋}) ∈ V)
49 fiuni 9468 . . . 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 2787 . 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 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wral 3061  wrex 3070  Vcvv 3480  cdif 3948  cun 3949  cin 3950  wss 3951  𝒫 cpw 4600  {csn 4626   cuni 4907   ciun 4991  cmpt 5225   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  cima 5688   Fn wfn 6556  wf 6557  cfv 6561  cmpo 7433  Xcixp 8937  Fincfn 8985  ficfi 9450  cardccrd 9975  topGenctg 17482  tcpt 17483  Topctop 22899  TopBasesctb 22952  Compccmp 23394  UFLcufl 23908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-2o 8507  df-ixp 8938  df-en 8986  df-dom 8987  df-fin 8989  df-fi 9451  df-topgen 17488  df-pt 17489  df-top 22900  df-bases 22953  df-cmp 23395
This theorem is referenced by:  ptcmplem5  24064
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