MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ptcmplem5 Structured version   Visualization version   GIF version

Theorem ptcmplem5 22594
Description: Lemma for ptcmp 22596. (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
ptcmplem5 (𝜑 → (∏t𝐹) ∈ Comp)
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 ptcmplem5
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptcmp.5 . . 3 (𝜑𝑋 ∈ (UFL ∩ dom card))
21elin1d 4174 . 2 (𝜑𝑋 ∈ UFL)
3 ptcmp.1 . . . 4 𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
4 ptcmp.2 . . . 4 𝑋 = X𝑛𝐴 (𝐹𝑛)
5 ptcmp.3 . . . 4 (𝜑𝐴𝑉)
6 ptcmp.4 . . . 4 (𝜑𝐹:𝐴⟶Comp)
73, 4, 5, 6, 1ptcmplem1 22590 . . 3 (𝜑 → (𝑋 = (ran 𝑆 ∪ {𝑋}) ∧ (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋})))))
87simpld 495 . 2 (𝜑𝑋 = (ran 𝑆 ∪ {𝑋}))
97simprd 496 . 2 (𝜑 → (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
10 elpwi 4549 . . . . . 6 (𝑦 ∈ 𝒫 ran 𝑆𝑦 ⊆ ran 𝑆)
115ad2antrr 722 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → 𝐴𝑉)
126ad2antrr 722 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → 𝐹:𝐴⟶Comp)
131ad2antrr 722 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → 𝑋 ∈ (UFL ∩ dom card))
14 simplrl 773 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → 𝑦 ⊆ ran 𝑆)
15 simplrr 774 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → 𝑋 = 𝑦)
16 simpr 485 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)
17 imaeq2 5919 . . . . . . . . . . 11 (𝑧 = 𝑢 → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑧) = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
1817eleq1d 2897 . . . . . . . . . 10 (𝑧 = 𝑢 → (((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑧) ∈ 𝑦 ↔ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑦))
1918cbvrabv 3492 . . . . . . . . 9 {𝑧 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑧) ∈ 𝑦} = {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑦}
203, 4, 11, 12, 13, 14, 15, 16, 19ptcmplem4 22593 . . . . . . . 8 ¬ ((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)
21 iman 402 . . . . . . . 8 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) ↔ ¬ ((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
2220, 21mpbir 232 . . . . . . 7 ((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)
2322expr 457 . . . . . 6 ((𝜑𝑦 ⊆ ran 𝑆) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
2410, 23sylan2 592 . . . . 5 ((𝜑𝑦 ∈ 𝒫 ran 𝑆) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
2524adantlr 711 . . . 4 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑦 ∈ 𝒫 ran 𝑆) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
26 velpw 4545 . . . . . . 7 (𝑦 ∈ 𝒫 (ran 𝑆 ∪ {𝑋}) ↔ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋}))
27 eldif 3945 . . . . . . . 8 (𝑦 ∈ (𝒫 (ran 𝑆 ∪ {𝑋}) ∖ 𝒫 ran 𝑆) ↔ (𝑦 ∈ 𝒫 (ran 𝑆 ∪ {𝑋}) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆))
28 elpwunsn 4615 . . . . . . . 8 (𝑦 ∈ (𝒫 (ran 𝑆 ∪ {𝑋}) ∖ 𝒫 ran 𝑆) → 𝑋𝑦)
2927, 28sylbir 236 . . . . . . 7 ((𝑦 ∈ 𝒫 (ran 𝑆 ∪ {𝑋}) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆) → 𝑋𝑦)
3026, 29sylanbr 582 . . . . . 6 ((𝑦 ⊆ (ran 𝑆 ∪ {𝑋}) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆) → 𝑋𝑦)
3130adantll 710 . . . . 5 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆) → 𝑋𝑦)
32 snssi 4735 . . . . . . . . 9 (𝑋𝑦 → {𝑋} ⊆ 𝑦)
3332adantl 482 . . . . . . . 8 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋𝑦) → {𝑋} ⊆ 𝑦)
34 snfi 8583 . . . . . . . 8 {𝑋} ∈ Fin
35 elfpw 8815 . . . . . . . 8 ({𝑋} ∈ (𝒫 𝑦 ∩ Fin) ↔ ({𝑋} ⊆ 𝑦 ∧ {𝑋} ∈ Fin))
3633, 34, 35sylanblrc 590 . . . . . . 7 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋𝑦) → {𝑋} ∈ (𝒫 𝑦 ∩ Fin))
37 unisng 4847 . . . . . . . . 9 (𝑋𝑦 {𝑋} = 𝑋)
3837eqcomd 2827 . . . . . . . 8 (𝑋𝑦𝑋 = {𝑋})
3938adantl 482 . . . . . . 7 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋𝑦) → 𝑋 = {𝑋})
40 unieq 4840 . . . . . . . 8 (𝑧 = {𝑋} → 𝑧 = {𝑋})
4140rspceeqv 3637 . . . . . . 7 (({𝑋} ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝑋 = {𝑋}) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)
4236, 39, 41syl2anc 584 . . . . . 6 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋𝑦) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)
4342a1d 25 . . . . 5 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋𝑦) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
4431, 43syldan 591 . . . 4 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
4525, 44pm2.61dan 809 . . 3 ((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
4645impr 455 . 2 ((𝜑 ∧ (𝑦 ⊆ (ran 𝑆 ∪ {𝑋}) ∧ 𝑋 = 𝑦)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)
472, 8, 9, 46alexsub 22583 1 (𝜑 → (∏t𝐹) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  wrex 3139  {crab 3142  cdif 3932  cun 3933  cin 3934  wss 3935  𝒫 cpw 4537  {csn 4559   cuni 4832  cmpt 5138  ccnv 5548  dom cdm 5549  ran crn 5550  cima 5552  wf 6345  cfv 6349  cmpo 7147  Xcixp 8450  Fincfn 8498  ficfi 8863  cardccrd 9353  topGenctg 16701  tcpt 16702  Compccmp 21924  UFLcufl 22438
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 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  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 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-iin 4915  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-2o 8094  df-oadd 8097  df-omul 8098  df-er 8279  df-map 8398  df-ixp 8451  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-fi 8864  df-wdom 9012  df-card 9357  df-acn 9360  df-topgen 16707  df-pt 16708  df-fbas 20472  df-fg 20473  df-top 21432  df-topon 21449  df-bases 21484  df-cld 21557  df-ntr 21558  df-cls 21559  df-nei 21636  df-cmp 21925  df-fil 22384  df-ufil 22439  df-ufl 22440  df-flim 22477  df-fcls 22479
This theorem is referenced by:  ptcmpg  22595
  Copyright terms: Public domain W3C validator