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Theorem ttukeylem2 10533
Description: Lemma for ttukey 10541. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.)
Hypotheses
Ref Expression
ttukeylem.1 (πœ‘ β†’ 𝐹:(cardβ€˜(βˆͺ 𝐴 βˆ– 𝐡))–1-1-ontoβ†’(βˆͺ 𝐴 βˆ– 𝐡))
ttukeylem.2 (πœ‘ β†’ 𝐡 ∈ 𝐴)
ttukeylem.3 (πœ‘ β†’ βˆ€π‘₯(π‘₯ ∈ 𝐴 ↔ (𝒫 π‘₯ ∩ Fin) βŠ† 𝐴))
Assertion
Ref Expression
ttukeylem2 ((πœ‘ ∧ (𝐢 ∈ 𝐴 ∧ 𝐷 βŠ† 𝐢)) β†’ 𝐷 ∈ 𝐴)
Distinct variable groups:   π‘₯,𝐢   π‘₯,𝐷   π‘₯,𝐴   π‘₯,𝐡   π‘₯,𝐹
Allowed substitution hint:   πœ‘(π‘₯)

Proof of Theorem ttukeylem2
StepHypRef Expression
1 simpr 483 . . . . . 6 ((πœ‘ ∧ 𝐷 βŠ† 𝐢) β†’ 𝐷 βŠ† 𝐢)
21sspwd 4611 . . . . 5 ((πœ‘ ∧ 𝐷 βŠ† 𝐢) β†’ 𝒫 𝐷 βŠ† 𝒫 𝐢)
3 ssrin 4228 . . . . 5 (𝒫 𝐷 βŠ† 𝒫 𝐢 β†’ (𝒫 𝐷 ∩ Fin) βŠ† (𝒫 𝐢 ∩ Fin))
4 sstr2 3979 . . . . 5 ((𝒫 𝐷 ∩ Fin) βŠ† (𝒫 𝐢 ∩ Fin) β†’ ((𝒫 𝐢 ∩ Fin) βŠ† 𝐴 β†’ (𝒫 𝐷 ∩ Fin) βŠ† 𝐴))
52, 3, 43syl 18 . . . 4 ((πœ‘ ∧ 𝐷 βŠ† 𝐢) β†’ ((𝒫 𝐢 ∩ Fin) βŠ† 𝐴 β†’ (𝒫 𝐷 ∩ Fin) βŠ† 𝐴))
6 ttukeylem.1 . . . . . 6 (πœ‘ β†’ 𝐹:(cardβ€˜(βˆͺ 𝐴 βˆ– 𝐡))–1-1-ontoβ†’(βˆͺ 𝐴 βˆ– 𝐡))
7 ttukeylem.2 . . . . . 6 (πœ‘ β†’ 𝐡 ∈ 𝐴)
8 ttukeylem.3 . . . . . 6 (πœ‘ β†’ βˆ€π‘₯(π‘₯ ∈ 𝐴 ↔ (𝒫 π‘₯ ∩ Fin) βŠ† 𝐴))
96, 7, 8ttukeylem1 10532 . . . . 5 (πœ‘ β†’ (𝐢 ∈ 𝐴 ↔ (𝒫 𝐢 ∩ Fin) βŠ† 𝐴))
109adantr 479 . . . 4 ((πœ‘ ∧ 𝐷 βŠ† 𝐢) β†’ (𝐢 ∈ 𝐴 ↔ (𝒫 𝐢 ∩ Fin) βŠ† 𝐴))
116, 7, 8ttukeylem1 10532 . . . . 5 (πœ‘ β†’ (𝐷 ∈ 𝐴 ↔ (𝒫 𝐷 ∩ Fin) βŠ† 𝐴))
1211adantr 479 . . . 4 ((πœ‘ ∧ 𝐷 βŠ† 𝐢) β†’ (𝐷 ∈ 𝐴 ↔ (𝒫 𝐷 ∩ Fin) βŠ† 𝐴))
135, 10, 123imtr4d 293 . . 3 ((πœ‘ ∧ 𝐷 βŠ† 𝐢) β†’ (𝐢 ∈ 𝐴 β†’ 𝐷 ∈ 𝐴))
1413impancom 450 . 2 ((πœ‘ ∧ 𝐢 ∈ 𝐴) β†’ (𝐷 βŠ† 𝐢 β†’ 𝐷 ∈ 𝐴))
1514impr 453 1 ((πœ‘ ∧ (𝐢 ∈ 𝐴 ∧ 𝐷 βŠ† 𝐢)) β†’ 𝐷 ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394  βˆ€wal 1531   ∈ wcel 2098   βˆ– cdif 3936   ∩ cin 3938   βŠ† wss 3939  π’« cpw 4598  βˆͺ cuni 4903  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  Fincfn 8962  cardccrd 9958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7869  df-1o 8485  df-en 8963  df-dom 8964  df-fin 8966
This theorem is referenced by:  ttukeylem6  10537  ttukeylem7  10538
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