Theorem ttukeylem2 9524

Description: Lemma for ttukey 9532. 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 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → 𝐷 ⊆ 𝐶)
2		sspwb 5066	. . . . . 6 ⊢ (𝐷 ⊆ 𝐶 ↔ 𝒫 𝐷 ⊆ 𝒫 𝐶)
3	1, 2	sylib 208	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → 𝒫 𝐷 ⊆ 𝒫 𝐶)
4		ssrin 3981	. . . . 5 ⊢ (𝒫 𝐷 ⊆ 𝒫 𝐶 → (𝒫 𝐷 ∩ Fin) ⊆ (𝒫 𝐶 ∩ Fin))
5		sstr2 3751	. . . . 5 ⊢ ((𝒫 𝐷 ∩ Fin) ⊆ (𝒫 𝐶 ∩ Fin) → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
6	3, 4, 5	3syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
7		ttukeylem.1	. . . . . 6 ⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
8		ttukeylem.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
9		ttukeylem.3	. . . . . 6 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
10	7, 8, 9	ttukeylem1 9523	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
11	10	adantr 472	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
12	7, 8, 9	ttukeylem1 9523	. . . . 5 ⊢ (𝜑 → (𝐷 ∈ 𝐴 ↔ (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
13	12	adantr 472	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → (𝐷 ∈ 𝐴 ↔ (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
14	6, 11, 13	3imtr4d 283	. . 3 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → (𝐶 ∈ 𝐴 → 𝐷 ∈ 𝐴))
15	14	impancom 455	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐷 ⊆ 𝐶 → 𝐷 ∈ 𝐴))
16	15	impr 650	1 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐶)) → 𝐷 ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1630   ∈ wcel 2139   ∖ cdif 3712   ∩ cin 3714   ⊆ wss 3715  𝒫 cpw 4302  ∪ cuni 4588  –1-1-onto→wf1o 6048  ‘cfv 6049  Fincfn 8121  cardccrd 8951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-1o 7729  df-en 8122  df-dom 8123  df-fin 8125

This theorem is referenced by:  ttukeylem6  9528  ttukeylem7  9529