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Theorem ptcmplem5 21854
Description: Lemma for ptcmp 21856. (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 inss1 3831 . . 3 (UFL ∩ dom card) ⊆ UFL
2 ptcmp.5 . . 3 (𝜑𝑋 ∈ (UFL ∩ dom card))
31, 2sseldi 3599 . 2 (𝜑𝑋 ∈ UFL)
4 ptcmp.1 . . . 4 𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
5 ptcmp.2 . . . 4 𝑋 = X𝑛𝐴 (𝐹𝑛)
6 ptcmp.3 . . . 4 (𝜑𝐴𝑉)
7 ptcmp.4 . . . 4 (𝜑𝐹:𝐴⟶Comp)
84, 5, 6, 7, 2ptcmplem1 21850 . . 3 (𝜑 → (𝑋 = (ran 𝑆 ∪ {𝑋}) ∧ (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋})))))
98simpld 475 . 2 (𝜑𝑋 = (ran 𝑆 ∪ {𝑋}))
108simprd 479 . 2 (𝜑 → (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋}))))
11 elpwi 4166 . . . . . 6 (𝑦 ∈ 𝒫 ran 𝑆𝑦 ⊆ ran 𝑆)
126ad2antrr 762 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → 𝐴𝑉)
137ad2antrr 762 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → 𝐹:𝐴⟶Comp)
142ad2antrr 762 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → 𝑋 ∈ (UFL ∩ dom card))
15 simplrl 800 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → 𝑦 ⊆ ran 𝑆)
16 simplrr 801 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → 𝑋 = 𝑦)
17 simpr 477 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) → ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)
18 imaeq2 5460 . . . . . . . . . . 11 (𝑧 = 𝑢 → ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑧) = ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))
1918eleq1d 2685 . . . . . . . . . 10 (𝑧 = 𝑢 → (((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑧) ∈ 𝑦 ↔ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑦))
2019cbvrabv 3197 . . . . . . . . 9 {𝑧 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑧) ∈ 𝑦} = {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑦}
214, 5, 12, 13, 14, 15, 16, 17, 20ptcmplem4 21853 . . . . . . . 8 ¬ ((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)
22 iman 440 . . . . . . . 8 (((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧) ↔ ¬ ((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) ∧ ¬ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
2321, 22mpbir 221 . . . . . . 7 ((𝜑 ∧ (𝑦 ⊆ ran 𝑆𝑋 = 𝑦)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)
2423expr 643 . . . . . 6 ((𝜑𝑦 ⊆ ran 𝑆) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
2511, 24sylan2 491 . . . . 5 ((𝜑𝑦 ∈ 𝒫 ran 𝑆) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
2625adantlr 751 . . . 4 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑦 ∈ 𝒫 ran 𝑆) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
27 selpw 4163 . . . . . . 7 (𝑦 ∈ 𝒫 (ran 𝑆 ∪ {𝑋}) ↔ 𝑦 ⊆ (ran 𝑆 ∪ {𝑋}))
28 eldif 3582 . . . . . . . 8 (𝑦 ∈ (𝒫 (ran 𝑆 ∪ {𝑋}) ∖ 𝒫 ran 𝑆) ↔ (𝑦 ∈ 𝒫 (ran 𝑆 ∪ {𝑋}) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆))
29 elpwunsn 4222 . . . . . . . 8 (𝑦 ∈ (𝒫 (ran 𝑆 ∪ {𝑋}) ∖ 𝒫 ran 𝑆) → 𝑋𝑦)
3028, 29sylbir 225 . . . . . . 7 ((𝑦 ∈ 𝒫 (ran 𝑆 ∪ {𝑋}) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆) → 𝑋𝑦)
3127, 30sylanbr 490 . . . . . 6 ((𝑦 ⊆ (ran 𝑆 ∪ {𝑋}) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆) → 𝑋𝑦)
3231adantll 750 . . . . 5 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆) → 𝑋𝑦)
33 snssi 4337 . . . . . . . . 9 (𝑋𝑦 → {𝑋} ⊆ 𝑦)
3433adantl 482 . . . . . . . 8 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋𝑦) → {𝑋} ⊆ 𝑦)
35 snfi 8035 . . . . . . . . 9 {𝑋} ∈ Fin
3635a1i 11 . . . . . . . 8 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋𝑦) → {𝑋} ∈ Fin)
37 elfpw 8265 . . . . . . . 8 ({𝑋} ∈ (𝒫 𝑦 ∩ Fin) ↔ ({𝑋} ⊆ 𝑦 ∧ {𝑋} ∈ Fin))
3834, 36, 37sylanbrc 698 . . . . . . 7 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋𝑦) → {𝑋} ∈ (𝒫 𝑦 ∩ Fin))
39 unisng 4450 . . . . . . . . 9 (𝑋𝑦 {𝑋} = 𝑋)
4039eqcomd 2627 . . . . . . . 8 (𝑋𝑦𝑋 = {𝑋})
4140adantl 482 . . . . . . 7 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋𝑦) → 𝑋 = {𝑋})
42 unieq 4442 . . . . . . . . 9 (𝑧 = {𝑋} → 𝑧 = {𝑋})
4342eqeq2d 2631 . . . . . . . 8 (𝑧 = {𝑋} → (𝑋 = 𝑧𝑋 = {𝑋}))
4443rspcev 3307 . . . . . . 7 (({𝑋} ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝑋 = {𝑋}) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)
4538, 41, 44syl2anc 693 . . . . . 6 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋𝑦) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)
4645a1d 25 . . . . 5 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ 𝑋𝑦) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
4732, 46syldan 487 . . . 4 (((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) ∧ ¬ 𝑦 ∈ 𝒫 ran 𝑆) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
4826, 47pm2.61dan 832 . . 3 ((𝜑𝑦 ⊆ (ran 𝑆 ∪ {𝑋})) → (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
4948impr 649 . 2 ((𝜑 ∧ (𝑦 ⊆ (ran 𝑆 ∪ {𝑋}) ∧ 𝑋 = 𝑦)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)
503, 9, 10, 49alexsub 21843 1 (𝜑 → (∏t𝐹) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1482  wcel 1989  wrex 2912  {crab 2915  cdif 3569  cun 3570  cin 3571  wss 3572  𝒫 cpw 4156  {csn 4175   cuni 4434  cmpt 4727  ccnv 5111  dom cdm 5112  ran crn 5113  cima 5115  wf 5882  cfv 5886  cmpt2 6649  Xcixp 7905  Fincfn 7952  ficfi 8313  cardccrd 8758  topGenctg 16092  tcpt 16093  Compccmp 21183  UFLcufl 21698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-iin 4521  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-se 5072  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-2o 7558  df-oadd 7561  df-omul 7562  df-er 7739  df-map 7856  df-ixp 7906  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-fi 8314  df-wdom 8461  df-card 8762  df-acn 8765  df-topgen 16098  df-pt 16099  df-fbas 19737  df-fg 19738  df-top 20693  df-topon 20710  df-bases 20744  df-cld 20817  df-ntr 20818  df-cls 20819  df-nei 20896  df-cmp 21184  df-fil 21644  df-ufil 21699  df-ufl 21700  df-flim 21737  df-fcls 21739
This theorem is referenced by:  ptcmpg  21855
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