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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
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