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Theorem ptcmplem1 22184
Description: Lemma for ptcmp 22190. (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 6257 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
4 eqid 2799 . . . . . . . 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 21702 . . . . . . 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 580 . . . . . 6 (𝜑 → (∏t𝐹) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
7 cmptop 21527 . . . . . . . . . . 11 (𝑥 ∈ Comp → 𝑥 ∈ Top)
87ssriv 3802 . . . . . . . . . 10 Comp ⊆ Top
9 fss 6269 . . . . . . . . . 10 ((𝐹:𝐴⟶Comp ∧ Comp ⊆ Top) → 𝐹:𝐴⟶Top)
102, 8, 9sylancl 581 . . . . . . . . 9 (𝜑𝐹:𝐴⟶Top)
11 ptcmp.2 . . . . . . . . . 10 𝑋 = X𝑛𝐴 (𝐹𝑛)
124, 11ptbasfi 21713 . . . . . . . . 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 580 . . . . . . . 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 3955 . . . . . . . . . 10 (ran 𝑆 ∪ {𝑋}) = ({𝑋} ∪ ran 𝑆)
15 ptcmp.1 . . . . . . . . . . . 12 𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
1615rneqi 5555 . . . . . . . . . . 11 ran 𝑆 = ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
1716uneq2i 3962 . . . . . . . . . 10 ({𝑋} ∪ ran 𝑆) = ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
1814, 17eqtri 2821 . . . . . . . . 9 (ran 𝑆 ∪ {𝑋}) = ({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢)))
1918fveq2i 6414 . . . . . . . 8 (fi‘(ran 𝑆 ∪ {𝑋})) = (fi‘({𝑋} ∪ ran (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))))
2013, 19syl6eqr 2851 . . . . . . 7 (𝜑 → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} = (fi‘(ran 𝑆 ∪ {𝑋})))
2120fveq2d 6415 . . . . . 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 2833 . . . . 5 (𝜑 → (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
2322unieqd 4638 . . . 4 (𝜑 (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
24 fibas 21110 . . . . 5 (fi‘(ran 𝑆 ∪ {𝑋})) ∈ TopBases
25 unitg 21100 . . . . 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 2849 . . 3 (𝜑 (∏t𝐹) = (fi‘(ran 𝑆 ∪ {𝑋})))
28 eqid 2799 . . . . . 6 (∏t𝐹) = (∏t𝐹)
2928ptuni 21726 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑛𝐴 (𝐹𝑛) = (∏t𝐹))
301, 10, 29syl2anc 580 . . . 4 (𝜑X𝑛𝐴 (𝐹𝑛) = (∏t𝐹))
3111, 30syl5eq 2845 . . 3 (𝜑𝑋 = (∏t𝐹))
32 ptcmp.5 . . . . . . 7 (𝜑𝑋 ∈ (UFL ∩ dom card))
3332pwexd 5049 . . . . . 6 (𝜑 → 𝒫 𝑋 ∈ V)
34 eqid 2799 . . . . . . . . . . . 12 (𝑤𝑋 ↦ (𝑤𝑘)) = (𝑤𝑋 ↦ (𝑤𝑘))
3534mptpreima 5847 . . . . . . . . . . 11 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) = {𝑤𝑋 ∣ (𝑤𝑘) ∈ 𝑢}
36 ssrab2 3883 . . . . . . . . . . 11 {𝑤𝑋 ∣ (𝑤𝑘) ∈ 𝑢} ⊆ 𝑋
3735, 36eqsstri 3831 . . . . . . . . . 10 ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ⊆ 𝑋
3832adantr 473 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘𝐴𝑢 ∈ (𝐹𝑘))) → 𝑋 ∈ (UFL ∩ dom card))
39 elpw2g 5019 . . . . . . . . . . 11 (𝑋 ∈ (UFL ∩ dom card) → (((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋 ↔ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ⊆ 𝑋))
4038, 39syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑘𝐴𝑢 ∈ (𝐹𝑘))) → (((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋 ↔ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ⊆ 𝑋))
4137, 40mpbiri 250 . . . . . . . . 9 ((𝜑 ∧ (𝑘𝐴𝑢 ∈ (𝐹𝑘))) → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋)
4241ralrimivva 3152 . . . . . . . 8 (𝜑 → ∀𝑘𝐴𝑢 ∈ (𝐹𝑘)((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋)
4315fmpt2x 7472 . . . . . . . 8 (∀𝑘𝐴𝑢 ∈ (𝐹𝑘)((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝒫 𝑋𝑆: 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝒫 𝑋)
4442, 43sylib 210 . . . . . . 7 (𝜑𝑆: 𝑘𝐴 ({𝑘} × (𝐹𝑘))⟶𝒫 𝑋)
4544frnd 6263 . . . . . 6 (𝜑 → ran 𝑆 ⊆ 𝒫 𝑋)
4633, 45ssexd 5000 . . . . 5 (𝜑 → ran 𝑆 ∈ V)
47 snex 5099 . . . . 5 {𝑋} ∈ V
48 unexg 7193 . . . . 5 ((ran 𝑆 ∈ V ∧ {𝑋} ∈ V) → (ran 𝑆 ∪ {𝑋}) ∈ V)
4946, 47, 48sylancl 581 . . . 4 (𝜑 → (ran 𝑆 ∪ {𝑋}) ∈ V)
50 fiuni 8576 . . . 4 ((ran 𝑆 ∪ {𝑋}) ∈ V → (ran 𝑆 ∪ {𝑋}) = (fi‘(ran 𝑆 ∪ {𝑋})))
5149, 50syl 17 . . 3 (𝜑 (ran 𝑆 ∪ {𝑋}) = (fi‘(ran 𝑆 ∪ {𝑋})))
5227, 31, 513eqtr4d 2843 . 2 (𝜑𝑋 = (ran 𝑆 ∪ {𝑋}))
5352, 22jca 508 1 (𝜑 → (𝑋 = (ran 𝑆 ∪ {𝑋}) ∧ (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋})))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wex 1875  wcel 2157  {cab 2785  wral 3089  wrex 3090  {crab 3093  Vcvv 3385  cdif 3766  cun 3767  cin 3768  wss 3769  𝒫 cpw 4349  {csn 4368   cuni 4628   ciun 4710  cmpt 4922   × cxp 5310  ccnv 5311  dom cdm 5312  ran crn 5313  cima 5315   Fn wfn 6096  wf 6097  cfv 6101  cmpt2 6880  Xcixp 8148  Fincfn 8195  ficfi 8558  cardccrd 9047  topGenctg 16413  tcpt 16414  Topctop 21026  TopBasesctb 21078  Compccmp 21518  UFLcufl 22032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-iin 4713  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-ixp 8149  df-en 8196  df-dom 8197  df-fin 8199  df-fi 8559  df-topgen 16419  df-pt 16420  df-top 21027  df-bases 21079  df-cmp 21519
This theorem is referenced by:  ptcmplem5  22188
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