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Theorem ttukeylem1 9369
Description: Lemma for ttukey 9378. Expand out the property of being an element of a property of finite character. (Contributed by Mario Carneiro, 15-May-2015.)
Hypotheses
Ref Expression
ttukeylem.1 (𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
ttukeylem.2 (𝜑𝐵𝐴)
ttukeylem.3 (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
Assertion
Ref Expression
ttukeylem1 (𝜑 → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
Distinct variable groups:   𝑥,𝐶   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ttukeylem1
StepHypRef Expression
1 elex 3243 . . 3 (𝐶𝐴𝐶 ∈ V)
21a1i 11 . 2 (𝜑 → (𝐶𝐴𝐶 ∈ V))
3 id 22 . . . . 5 ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)
4 ssun1 3809 . . . . . . . 8 𝐴 ⊆ ( 𝐴𝐵)
5 undif1 4076 . . . . . . . 8 (( 𝐴𝐵) ∪ 𝐵) = ( 𝐴𝐵)
64, 5sseqtr4i 3671 . . . . . . 7 𝐴 ⊆ (( 𝐴𝐵) ∪ 𝐵)
7 fvex 6239 . . . . . . . . 9 (card‘( 𝐴𝐵)) ∈ V
8 ttukeylem.1 . . . . . . . . . 10 (𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
9 f1ofo 6182 . . . . . . . . . 10 (𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) → 𝐹:(card‘( 𝐴𝐵))–onto→( 𝐴𝐵))
108, 9syl 17 . . . . . . . . 9 (𝜑𝐹:(card‘( 𝐴𝐵))–onto→( 𝐴𝐵))
11 fornex 7177 . . . . . . . . 9 ((card‘( 𝐴𝐵)) ∈ V → (𝐹:(card‘( 𝐴𝐵))–onto→( 𝐴𝐵) → ( 𝐴𝐵) ∈ V))
127, 10, 11mpsyl 68 . . . . . . . 8 (𝜑 → ( 𝐴𝐵) ∈ V)
13 ttukeylem.2 . . . . . . . 8 (𝜑𝐵𝐴)
14 unexg 7001 . . . . . . . 8 ((( 𝐴𝐵) ∈ V ∧ 𝐵𝐴) → (( 𝐴𝐵) ∪ 𝐵) ∈ V)
1512, 13, 14syl2anc 694 . . . . . . 7 (𝜑 → (( 𝐴𝐵) ∪ 𝐵) ∈ V)
16 ssexg 4837 . . . . . . 7 (( 𝐴 ⊆ (( 𝐴𝐵) ∪ 𝐵) ∧ (( 𝐴𝐵) ∪ 𝐵) ∈ V) → 𝐴 ∈ V)
176, 15, 16sylancr 696 . . . . . 6 (𝜑 𝐴 ∈ V)
18 uniexb 7015 . . . . . 6 (𝐴 ∈ V ↔ 𝐴 ∈ V)
1917, 18sylibr 224 . . . . 5 (𝜑𝐴 ∈ V)
20 ssexg 4837 . . . . 5 (((𝒫 𝐶 ∩ Fin) ⊆ 𝐴𝐴 ∈ V) → (𝒫 𝐶 ∩ Fin) ∈ V)
213, 19, 20syl2anr 494 . . . 4 ((𝜑 ∧ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴) → (𝒫 𝐶 ∩ Fin) ∈ V)
22 infpwfidom 8889 . . . 4 ((𝒫 𝐶 ∩ Fin) ∈ V → 𝐶 ≼ (𝒫 𝐶 ∩ Fin))
23 reldom 8003 . . . . 5 Rel ≼
2423brrelexi 5192 . . . 4 (𝐶 ≼ (𝒫 𝐶 ∩ Fin) → 𝐶 ∈ V)
2521, 22, 243syl 18 . . 3 ((𝜑 ∧ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴) → 𝐶 ∈ V)
2625ex 449 . 2 (𝜑 → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴𝐶 ∈ V))
27 ttukeylem.3 . . 3 (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
28 eleq1 2718 . . . . 5 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
29 pweq 4194 . . . . . . 7 (𝑥 = 𝐶 → 𝒫 𝑥 = 𝒫 𝐶)
3029ineq1d 3846 . . . . . 6 (𝑥 = 𝐶 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝐶 ∩ Fin))
3130sseq1d 3665 . . . . 5 (𝑥 = 𝐶 → ((𝒫 𝑥 ∩ Fin) ⊆ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
3228, 31bibi12d 334 . . . 4 (𝑥 = 𝐶 → ((𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴) ↔ (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)))
3332spcgv 3324 . . 3 (𝐶 ∈ V → (∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴) → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)))
3427, 33syl5com 31 . 2 (𝜑 → (𝐶 ∈ V → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)))
352, 26, 34pm5.21ndd 368 1 (𝜑 → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wcel 2030  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  𝒫 cpw 4191   cuni 4468   class class class wbr 4685  ontowfo 5924  1-1-ontowf1o 5925  cfv 5926  cdom 7995  Fincfn 7997  cardccrd 8799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-en 7998  df-dom 7999  df-fin 8001
This theorem is referenced by:  ttukeylem2  9370  ttukeylem6  9374
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