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Theorem ttukeylem2 9936
 Description: Lemma for ttukey 9944. 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 488 . . . . . 6 ((𝜑𝐷𝐶) → 𝐷𝐶)
21sspwd 4514 . . . . 5 ((𝜑𝐷𝐶) → 𝒫 𝐷 ⊆ 𝒫 𝐶)
3 ssrin 4162 . . . . 5 (𝒫 𝐷 ⊆ 𝒫 𝐶 → (𝒫 𝐷 ∩ Fin) ⊆ (𝒫 𝐶 ∩ Fin))
4 sstr2 3923 . . . . 5 ((𝒫 𝐷 ∩ Fin) ⊆ (𝒫 𝐶 ∩ Fin) → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
52, 3, 43syl 18 . . . 4 ((𝜑𝐷𝐶) → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
6 ttukeylem.1 . . . . . 6 (𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
7 ttukeylem.2 . . . . . 6 (𝜑𝐵𝐴)
8 ttukeylem.3 . . . . . 6 (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
96, 7, 8ttukeylem1 9935 . . . . 5 (𝜑 → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
109adantr 484 . . . 4 ((𝜑𝐷𝐶) → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
116, 7, 8ttukeylem1 9935 . . . . 5 (𝜑 → (𝐷𝐴 ↔ (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
1211adantr 484 . . . 4 ((𝜑𝐷𝐶) → (𝐷𝐴 ↔ (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
135, 10, 123imtr4d 297 . . 3 ((𝜑𝐷𝐶) → (𝐶𝐴𝐷𝐴))
1413impancom 455 . 2 ((𝜑𝐶𝐴) → (𝐷𝐶𝐷𝐴))
1514impr 458 1 ((𝜑 ∧ (𝐶𝐴𝐷𝐶)) → 𝐷𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   ∈ wcel 2111   ∖ cdif 3879   ∩ cin 3881   ⊆ wss 3882  𝒫 cpw 4499  ∪ cuni 4803  –1-1-onto→wf1o 6328  ‘cfv 6329  Fincfn 8507  cardccrd 9363 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7451 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3722  df-csb 3830  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-pss 3901  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-om 7571  df-1o 8100  df-en 8508  df-dom 8509  df-fin 8511 This theorem is referenced by:  ttukeylem6  9940  ttukeylem7  9941
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