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Theorem ttukeylem1 9275
Description: Lemma for ttukey 9284. 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 3198 . . 3 (𝐶𝐴𝐶 ∈ V)
21a1i 11 . 2 (𝜑 → (𝐶𝐴𝐶 ∈ V))
3 id 22 . . . . 5 ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)
4 ssun1 3754 . . . . . . . 8 𝐴 ⊆ ( 𝐴𝐵)
5 undif1 4015 . . . . . . . 8 (( 𝐴𝐵) ∪ 𝐵) = ( 𝐴𝐵)
64, 5sseqtr4i 3617 . . . . . . 7 𝐴 ⊆ (( 𝐴𝐵) ∪ 𝐵)
7 fvex 6158 . . . . . . . . 9 (card‘( 𝐴𝐵)) ∈ V
8 ttukeylem.1 . . . . . . . . . 10 (𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
9 f1ofo 6101 . . . . . . . . . 10 (𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) → 𝐹:(card‘( 𝐴𝐵))–onto→( 𝐴𝐵))
108, 9syl 17 . . . . . . . . 9 (𝜑𝐹:(card‘( 𝐴𝐵))–onto→( 𝐴𝐵))
11 fornex 7082 . . . . . . . . 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 6912 . . . . . . . 8 ((( 𝐴𝐵) ∈ V ∧ 𝐵𝐴) → (( 𝐴𝐵) ∪ 𝐵) ∈ V)
1512, 13, 14syl2anc 692 . . . . . . 7 (𝜑 → (( 𝐴𝐵) ∪ 𝐵) ∈ V)
16 ssexg 4764 . . . . . . 7 (( 𝐴 ⊆ (( 𝐴𝐵) ∪ 𝐵) ∧ (( 𝐴𝐵) ∪ 𝐵) ∈ V) → 𝐴 ∈ V)
176, 15, 16sylancr 694 . . . . . 6 (𝜑 𝐴 ∈ V)
18 uniexb 6921 . . . . . 6 (𝐴 ∈ V ↔ 𝐴 ∈ V)
1917, 18sylibr 224 . . . . 5 (𝜑𝐴 ∈ V)
20 ssexg 4764 . . . . 5 (((𝒫 𝐶 ∩ Fin) ⊆ 𝐴𝐴 ∈ V) → (𝒫 𝐶 ∩ Fin) ∈ V)
213, 19, 20syl2anr 495 . . . 4 ((𝜑 ∧ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴) → (𝒫 𝐶 ∩ Fin) ∈ V)
22 infpwfidom 8795 . . . 4 ((𝒫 𝐶 ∩ Fin) ∈ V → 𝐶 ≼ (𝒫 𝐶 ∩ Fin))
23 reldom 7905 . . . . 5 Rel ≼
2423brrelexi 5118 . . . 4 (𝐶 ≼ (𝒫 𝐶 ∩ Fin) → 𝐶 ∈ V)
2521, 22, 243syl 18 . . 3 ((𝜑 ∧ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴) → 𝐶 ∈ V)
2625ex 450 . 2 (𝜑 → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴𝐶 ∈ V))
27 ttukeylem.3 . . 3 (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
28 eleq1 2686 . . . . 5 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
29 pweq 4133 . . . . . . 7 (𝑥 = 𝐶 → 𝒫 𝑥 = 𝒫 𝐶)
3029ineq1d 3791 . . . . . 6 (𝑥 = 𝐶 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝐶 ∩ Fin))
3130sseq1d 3611 . . . . 5 (𝑥 = 𝐶 → ((𝒫 𝑥 ∩ Fin) ⊆ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
3228, 31bibi12d 335 . . . 4 (𝑥 = 𝐶 → ((𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴) ↔ (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)))
3332spcgv 3279 . . 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 369 1 (𝜑 → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wcel 1987  Vcvv 3186  cdif 3552  cun 3553  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402   class class class wbr 4613  ontowfo 5845  1-1-ontowf1o 5846  cfv 5847  cdom 7897  Fincfn 7899  cardccrd 8705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-1o 7505  df-en 7900  df-dom 7901  df-fin 7903
This theorem is referenced by:  ttukeylem2  9276  ttukeylem6  9280
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