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Theorem ttukeylem2 10507
Description: Lemma for ttukey 10515. 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 484 . . . . . 6 ((πœ‘ ∧ 𝐷 βŠ† 𝐢) β†’ 𝐷 βŠ† 𝐢)
21sspwd 4610 . . . . 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 3984 . . . . 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 10506 . . . . 5 (πœ‘ β†’ (𝐢 ∈ 𝐴 ↔ (𝒫 𝐢 ∩ Fin) βŠ† 𝐴))
109adantr 480 . . . 4 ((πœ‘ ∧ 𝐷 βŠ† 𝐢) β†’ (𝐢 ∈ 𝐴 ↔ (𝒫 𝐢 ∩ Fin) βŠ† 𝐴))
116, 7, 8ttukeylem1 10506 . . . . 5 (πœ‘ β†’ (𝐷 ∈ 𝐴 ↔ (𝒫 𝐷 ∩ Fin) βŠ† 𝐴))
1211adantr 480 . . . 4 ((πœ‘ ∧ 𝐷 βŠ† 𝐢) β†’ (𝐷 ∈ 𝐴 ↔ (𝒫 𝐷 ∩ Fin) βŠ† 𝐴))
135, 10, 123imtr4d 294 . . 3 ((πœ‘ ∧ 𝐷 βŠ† 𝐢) β†’ (𝐢 ∈ 𝐴 β†’ 𝐷 ∈ 𝐴))
1413impancom 451 . 2 ((πœ‘ ∧ 𝐢 ∈ 𝐴) β†’ (𝐷 βŠ† 𝐢 β†’ 𝐷 ∈ 𝐴))
1514impr 454 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 395  βˆ€wal 1531   ∈ wcel 2098   βˆ– cdif 3940   ∩ cin 3942   βŠ† wss 3943  π’« cpw 4597  βˆͺ cuni 4902  β€“1-1-ontoβ†’wf1o 6536  β€˜cfv 6537  Fincfn 8941  cardccrd 9932
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7853  df-1o 8467  df-en 8942  df-dom 8943  df-fin 8945
This theorem is referenced by:  ttukeylem6  10511  ttukeylem7  10512
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