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Theorem ptcmplem5 24064
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
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 4204 . 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 24060 . . 3 (𝜑 → (𝑋 = (ran 𝑆 ∪ {𝑋}) ∧ (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋})))))
87simpld 494 . 2 (𝜑𝑋 = (ran 𝑆 ∪ {𝑋}))
97simprd 495 . 2 (𝜑 → (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
10 elpwi 4607 . . . . . 6 (𝑦 ∈ 𝒫 ran 𝑆𝑦 ⊆ ran 𝑆)
115ad2antrr 726 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → 𝐴𝑉)
126ad2antrr 726 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → 𝐹:𝐴⟶Comp)
131ad2antrr 726 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → 𝑋 ∈ (UFL ∩ dom card))
14 simplrl 777 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → 𝑦 ⊆ ran 𝑆)
15 simplrr 778 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → 𝑋 = 𝑦)
16 simpr 484 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)
17 imaeq2 6074 . . . . . . . . . . 11 (𝑧 = 𝑢 → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑧) = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
1817eleq1d 2826 . . . . . . . . . 10 (𝑧 = 𝑢 → (((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑧) ∈ 𝑦 ↔ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑦))
1918cbvrabv 3447 . . . . . . . . 9 {𝑧 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑧) ∈ 𝑦} = {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑦}
203, 4, 11, 12, 13, 14, 15, 16, 19ptcmplem4 24063 . . . . . . . 8 ¬ ((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)
21 iman 401 . . . . . . . 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 231 . . . . . . 7 ((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)
2322expr 456 . . . . . 6 ((𝜑𝑦 ⊆ ran 𝑆) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
2410, 23sylan2 593 . . . . 5 ((𝜑𝑦 ∈ 𝒫 ran 𝑆) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
2524adantlr 715 . . . 4 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑦 ∈ 𝒫 ran 𝑆) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
26 velpw 4605 . . . . . . 7 (𝑦 ∈ 𝒫 (ran 𝑆 ∪ {𝑋}) ↔ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋}))
27 eldif 3961 . . . . . . . 8 (𝑦 ∈ (𝒫 (ran 𝑆 ∪ {𝑋}) ∖ 𝒫 ran 𝑆) ↔ (𝑦 ∈ 𝒫 (ran 𝑆 ∪ {𝑋}) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆))
28 elpwunsn 4684 . . . . . . . 8 (𝑦 ∈ (𝒫 (ran 𝑆 ∪ {𝑋}) ∖ 𝒫 ran 𝑆) → 𝑋𝑦)
2927, 28sylbir 235 . . . . . . 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 714 . . . . 5 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆) → 𝑋𝑦)
32 snssi 4808 . . . . . . . . 9 (𝑋𝑦 → {𝑋} ⊆ 𝑦)
3332adantl 481 . . . . . . . 8 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋𝑦) → {𝑋} ⊆ 𝑦)
34 snfi 9083 . . . . . . . 8 {𝑋} ∈ Fin
35 elfpw 9394 . . . . . . . 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 4925 . . . . . . . . 9 (𝑋𝑦 {𝑋} = 𝑋)
3837eqcomd 2743 . . . . . . . 8 (𝑋𝑦𝑋 = {𝑋})
3938adantl 481 . . . . . . 7 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋𝑦) → 𝑋 = {𝑋})
40 unieq 4918 . . . . . . . 8 (𝑧 = {𝑋} → 𝑧 = {𝑋})
4140rspceeqv 3645 . . . . . . 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 813 . . 3 ((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
4645impr 454 . 2 ((𝜑 ∧ (𝑦 ⊆ (ran 𝑆 ∪ {𝑋}) ∧ 𝑋 = 𝑦)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)
472, 8, 9, 46alexsub 24053 1 (𝜑 → (∏t𝐹) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wrex 3070  {crab 3436  cdif 3948  cun 3949  cin 3950  wss 3951  𝒫 cpw 4600  {csn 4626   cuni 4907  cmpt 5225  ccnv 5684  dom cdm 5685  ran crn 5686  cima 5688  wf 6557  cfv 6561  cmpo 7433  Xcixp 8937  Fincfn 8985  ficfi 9450  cardccrd 9975  topGenctg 17482  tcpt 17483  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-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  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-se 5638  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-pred 6321  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-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-er 8745  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-fin 8989  df-fi 9451  df-wdom 9605  df-card 9979  df-acn 9982  df-topgen 17488  df-pt 17489  df-fbas 21361  df-fg 21362  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-cmp 23395  df-fil 23854  df-ufil 23909  df-ufl 23910  df-flim 23947  df-fcls 23949
This theorem is referenced by:  ptcmpg  24065
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