Theorem ttukeylem2 10550

Description: Lemma for ttukey 10558. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.)

Hypotheses

Ref	Expression
ttukeylem.1	⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
ttukeylem.2	⊢ (𝜑 → 𝐵 ∈ 𝐴)
ttukeylem.3	⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))

Assertion

Ref	Expression
ttukeylem2	⊢ ((𝜑 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐶)) → 𝐷 ∈ 𝐴)

Distinct variable groups:   𝑥,𝐶   𝑥,𝐷   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ttukeylem2

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → 𝐷 ⊆ 𝐶)
2	1	sspwd 4613	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → 𝒫 𝐷 ⊆ 𝒫 𝐶)
3		ssrin 4242	. . . . 5 ⊢ (𝒫 𝐷 ⊆ 𝒫 𝐶 → (𝒫 𝐷 ∩ Fin) ⊆ (𝒫 𝐶 ∩ Fin))
4		sstr2 3990	. . . . 5 ⊢ ((𝒫 𝐷 ∩ Fin) ⊆ (𝒫 𝐶 ∩ Fin) → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
5	2, 3, 4	3syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
6		ttukeylem.1	. . . . . 6 ⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
7		ttukeylem.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
8		ttukeylem.3	. . . . . 6 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
9	6, 7, 8	ttukeylem1 10549	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
10	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
11	6, 7, 8	ttukeylem1 10549	. . . . 5 ⊢ (𝜑 → (𝐷 ∈ 𝐴 ↔ (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
12	11	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → (𝐷 ∈ 𝐴 ↔ (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
13	5, 10, 12	3imtr4d 294	. . 3 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → (𝐶 ∈ 𝐴 → 𝐷 ∈ 𝐴))
14	13	impancom 451	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐷 ⊆ 𝐶 → 𝐷 ∈ 𝐴))
15	14	impr 454	1 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐶)) → 𝐷 ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   ∈ wcel 2108   ∖ cdif 3948   ∩ cin 3950   ⊆ wss 3951  𝒫 cpw 4600  ∪ cuni 4907  –1-1-onto→wf1o 6560  ‘cfv 6561  Fincfn 8985  cardccrd 9975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-en 8986  df-dom 8987  df-fin 8989

This theorem is referenced by:  ttukeylem6  10554  ttukeylem7  10555