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Theorem ttukeylem2 9524
Description: Lemma for ttukey 9532. 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 479 . . . . . 6 ((𝜑𝐷𝐶) → 𝐷𝐶)
2 sspwb 5066 . . . . . 6 (𝐷𝐶 ↔ 𝒫 𝐷 ⊆ 𝒫 𝐶)
31, 2sylib 208 . . . . 5 ((𝜑𝐷𝐶) → 𝒫 𝐷 ⊆ 𝒫 𝐶)
4 ssrin 3981 . . . . 5 (𝒫 𝐷 ⊆ 𝒫 𝐶 → (𝒫 𝐷 ∩ Fin) ⊆ (𝒫 𝐶 ∩ Fin))
5 sstr2 3751 . . . . 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 9523 . . . . 5 (𝜑 → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
1110adantr 472 . . . 4 ((𝜑𝐷𝐶) → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
127, 8, 9ttukeylem1 9523 . . . . 5 (𝜑 → (𝐷𝐴 ↔ (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
1312adantr 472 . . . 4 ((𝜑𝐷𝐶) → (𝐷𝐴 ↔ (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
146, 11, 133imtr4d 283 . . 3 ((𝜑𝐷𝐶) → (𝐶𝐴𝐷𝐴))
1514impancom 455 . 2 ((𝜑𝐶𝐴) → (𝐷𝐶𝐷𝐴))
1615impr 650 1 ((𝜑 ∧ (𝐶𝐴𝐷𝐶)) → 𝐷𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630  wcel 2139  cdif 3712  cin 3714  wss 3715  𝒫 cpw 4302   cuni 4588  1-1-ontowf1o 6048  cfv 6049  Fincfn 8121  cardccrd 8951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-1o 7729  df-en 8122  df-dom 8123  df-fin 8125
This theorem is referenced by:  ttukeylem6  9528  ttukeylem7  9529
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