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Theorem ttukeylem2 10447
Description: Lemma for ttukey 10455. 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 486 . . . . . 6 ((πœ‘ ∧ 𝐷 βŠ† 𝐢) β†’ 𝐷 βŠ† 𝐢)
21sspwd 4574 . . . . 5 ((πœ‘ ∧ 𝐷 βŠ† 𝐢) β†’ 𝒫 𝐷 βŠ† 𝒫 𝐢)
3 ssrin 4194 . . . . 5 (𝒫 𝐷 βŠ† 𝒫 𝐢 β†’ (𝒫 𝐷 ∩ Fin) βŠ† (𝒫 𝐢 ∩ Fin))
4 sstr2 3952 . . . . 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 10446 . . . . 5 (πœ‘ β†’ (𝐢 ∈ 𝐴 ↔ (𝒫 𝐢 ∩ Fin) βŠ† 𝐴))
109adantr 482 . . . 4 ((πœ‘ ∧ 𝐷 βŠ† 𝐢) β†’ (𝐢 ∈ 𝐴 ↔ (𝒫 𝐢 ∩ Fin) βŠ† 𝐴))
116, 7, 8ttukeylem1 10446 . . . . 5 (πœ‘ β†’ (𝐷 ∈ 𝐴 ↔ (𝒫 𝐷 ∩ Fin) βŠ† 𝐴))
1211adantr 482 . . . 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 453 . 2 ((πœ‘ ∧ 𝐢 ∈ 𝐴) β†’ (𝐷 βŠ† 𝐢 β†’ 𝐷 ∈ 𝐴))
1514impr 456 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 397  βˆ€wal 1540   ∈ wcel 2107   βˆ– cdif 3908   ∩ cin 3910   βŠ† wss 3911  π’« cpw 4561  βˆͺ cuni 4866  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  Fincfn 8884  cardccrd 9872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-om 7804  df-1o 8413  df-en 8885  df-dom 8886  df-fin 8888
This theorem is referenced by:  ttukeylem6  10451  ttukeylem7  10452
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