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Theorem ttukeylem2 10446
Description: Lemma for ttukey 10454. 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 485 . . . . . 6 ((𝜑𝐷𝐶) → 𝐷𝐶)
21sspwd 4573 . . . . 5 ((𝜑𝐷𝐶) → 𝒫 𝐷 ⊆ 𝒫 𝐶)
3 ssrin 4193 . . . . 5 (𝒫 𝐷 ⊆ 𝒫 𝐶 → (𝒫 𝐷 ∩ Fin) ⊆ (𝒫 𝐶 ∩ Fin))
4 sstr2 3951 . . . . 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 10445 . . . . 5 (𝜑 → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
109adantr 481 . . . 4 ((𝜑𝐷𝐶) → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
116, 7, 8ttukeylem1 10445 . . . . 5 (𝜑 → (𝐷𝐴 ↔ (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
1211adantr 481 . . . 4 ((𝜑𝐷𝐶) → (𝐷𝐴 ↔ (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
135, 10, 123imtr4d 293 . . 3 ((𝜑𝐷𝐶) → (𝐶𝐴𝐷𝐴))
1413impancom 452 . 2 ((𝜑𝐶𝐴) → (𝐷𝐶𝐷𝐴))
1514impr 455 1 ((𝜑 ∧ (𝐶𝐴𝐷𝐶)) → 𝐷𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539  wcel 2106  cdif 3907  cin 3909  wss 3910  𝒫 cpw 4560   cuni 4865  1-1-ontowf1o 6495  cfv 6496  Fincfn 8883  cardccrd 9871
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-1o 8412  df-en 8884  df-dom 8885  df-fin 8887
This theorem is referenced by:  ttukeylem6  10450  ttukeylem7  10451
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