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Theorem ttukeylem2 9276
Description: Lemma for ttukey 9284. A property of finite character is closed under subsets. (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
ttukeylem2 ((𝜑 ∧ (𝐶𝐴𝐷𝐶)) → 𝐷𝐴)
Distinct variable groups:   𝑥,𝐶   𝑥,𝐷   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ttukeylem2
StepHypRef Expression
1 simpr 477 . . . . . 6 ((𝜑𝐷𝐶) → 𝐷𝐶)
2 sspwb 4878 . . . . . 6 (𝐷𝐶 ↔ 𝒫 𝐷 ⊆ 𝒫 𝐶)
31, 2sylib 208 . . . . 5 ((𝜑𝐷𝐶) → 𝒫 𝐷 ⊆ 𝒫 𝐶)
4 ssrin 3816 . . . . 5 (𝒫 𝐷 ⊆ 𝒫 𝐶 → (𝒫 𝐷 ∩ Fin) ⊆ (𝒫 𝐶 ∩ Fin))
5 sstr2 3590 . . . . 5 ((𝒫 𝐷 ∩ Fin) ⊆ (𝒫 𝐶 ∩ Fin) → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
63, 4, 53syl 18 . . . 4 ((𝜑𝐷𝐶) → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
7 ttukeylem.1 . . . . . 6 (𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
8 ttukeylem.2 . . . . . 6 (𝜑𝐵𝐴)
9 ttukeylem.3 . . . . . 6 (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
107, 8, 9ttukeylem1 9275 . . . . 5 (𝜑 → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
1110adantr 481 . . . 4 ((𝜑𝐷𝐶) → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
127, 8, 9ttukeylem1 9275 . . . . 5 (𝜑 → (𝐷𝐴 ↔ (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
1312adantr 481 . . . 4 ((𝜑𝐷𝐶) → (𝐷𝐴 ↔ (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
146, 11, 133imtr4d 283 . . 3 ((𝜑𝐷𝐶) → (𝐶𝐴𝐷𝐴))
1514impancom 456 . 2 ((𝜑𝐶𝐴) → (𝐷𝐶𝐷𝐴))
1615impr 648 1 ((𝜑 ∧ (𝐶𝐴𝐷𝐶)) → 𝐷𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478  wcel 1987  cdif 3552  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402  1-1-ontowf1o 5846  cfv 5847  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:  ttukeylem6  9280  ttukeylem7  9281
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