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

Theorem ptcmplem5 22583
Description: Lemma for ptcmp 22585. (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 4179 . 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 22579 . . 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 4554 . . . . . 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 5923 . . . . . . . . . . 11 (𝑧 = 𝑢 → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑧) = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
1817eleq1d 2902 . . . . . . . . . 10 (𝑧 = 𝑢 → (((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑧) ∈ 𝑦 ↔ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑦))
1918cbvrabv 3497 . . . . . . . . 9 {𝑧 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑧) ∈ 𝑦} = {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑦}
203, 4, 11, 12, 13, 14, 15, 16, 19ptcmplem4 22582 . . . . . . . 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 4550 . . . . . . 7 (𝑦 ∈ 𝒫 (ran 𝑆 ∪ {𝑋}) ↔ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋}))
27 eldif 3950 . . . . . . . 8 (𝑦 ∈ (𝒫 (ran 𝑆 ∪ {𝑋}) ∖ 𝒫 ran 𝑆) ↔ (𝑦 ∈ 𝒫 (ran 𝑆 ∪ {𝑋}) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆))
28 elpwunsn 4620 . . . . . . . 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 4740 . . . . . . . . 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 4852 . . . . . . . . 9 (𝑋𝑦 {𝑋} = 𝑋)
3837eqcomd 2832 . . . . . . . 8 (𝑋𝑦𝑋 = {𝑋})
3938adantl 482 . . . . . . 7 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋𝑦) → 𝑋 = {𝑋})
40 unieq 4845 . . . . . . . 8 (𝑧 = {𝑋} → 𝑧 = {𝑋})
4140rspceeqv 3642 . . . . . . 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 22572 1 (𝜑 → (∏t𝐹) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1530  wcel 2107  wrex 3144  {crab 3147  cdif 3937  cun 3938  cin 3939  wss 3940  𝒫 cpw 4542  {csn 4564   cuni 4837  cmpt 5143  ccnv 5553  dom cdm 5554  ran crn 5555  cima 5557  wf 6348  cfv 6352  cmpo 7150  Xcixp 8450  Fincfn 8498  ficfi 8863  cardccrd 9353  topGenctg 16701  tcpt 16702  Compccmp 21913  UFLcufl 22427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  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 20461  df-fg 20462  df-top 21421  df-topon 21438  df-bases 21473  df-cld 21546  df-ntr 21547  df-cls 21548  df-nei 21625  df-cmp 21914  df-fil 22373  df-ufil 22428  df-ufl 22429  df-flim 22466  df-fcls 22468
This theorem is referenced by:  ptcmpg  22584
  Copyright terms: Public domain W3C validator