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Theorem ttukeylem2 9778
 Description: Lemma for ttukey 9786. 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)
2 sspwb 5233 . . . . . 6 (𝐷𝐶 ↔ 𝒫 𝐷 ⊆ 𝒫 𝐶)
31, 2sylib 219 . . . . 5 ((𝜑𝐷𝐶) → 𝒫 𝐷 ⊆ 𝒫 𝐶)
4 ssrin 4130 . . . . 5 (𝒫 𝐷 ⊆ 𝒫 𝐶 → (𝒫 𝐷 ∩ Fin) ⊆ (𝒫 𝐶 ∩ Fin))
5 sstr2 3896 . . . . 5 ((𝒫 𝐷 ∩ Fin) ⊆ (𝒫 𝐶 ∩ Fin) → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
63, 4, 53syl 18 . . . 4 ((𝜑𝐷𝐶) → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
7 ttukeylem.1 . . . . . 6 (𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
8 ttukeylem.2 . . . . . 6 (𝜑𝐵𝐴)
9 ttukeylem.3 . . . . . 6 (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
107, 8, 9ttukeylem1 9777 . . . . 5 (𝜑 → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
1110adantr 481 . . . 4 ((𝜑𝐷𝐶) → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
127, 8, 9ttukeylem1 9777 . . . . 5 (𝜑 → (𝐷𝐴 ↔ (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
1312adantr 481 . . . 4 ((𝜑𝐷𝐶) → (𝐷𝐴 ↔ (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
146, 11, 133imtr4d 295 . . 3 ((𝜑𝐷𝐶) → (𝐶𝐴𝐷𝐴))
1514impancom 452 . 2 ((𝜑𝐶𝐴) → (𝐷𝐶𝐷𝐴))
1615impr 455 1 ((𝜑 ∧ (𝐶𝐴𝐷𝐶)) → 𝐷𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1520   ∈ wcel 2081   ∖ cdif 3856   ∩ cin 3858   ⊆ wss 3859  𝒫 cpw 4453  ∪ cuni 4745  –1-1-onto→wf1o 6224  ‘cfv 6225  Fincfn 8357  cardccrd 9210 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-om 7437  df-1o 7953  df-en 8358  df-dom 8359  df-fin 8361 This theorem is referenced by:  ttukeylem6  9782  ttukeylem7  9783
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