Theorem ttukeylem2 9935

Description: Lemma for ttukey 9943. 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 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → 𝐷 ⊆ 𝐶)
2	1	sspwd 4557	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → 𝒫 𝐷 ⊆ 𝒫 𝐶)
3		ssrin 4213	. . . . 5 ⊢ (𝒫 𝐷 ⊆ 𝒫 𝐶 → (𝒫 𝐷 ∩ Fin) ⊆ (𝒫 𝐶 ∩ Fin))
4		sstr2 3977	. . . . 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 9934	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
10	9	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
11	6, 7, 8	ttukeylem1 9934	. . . . 5 ⊢ (𝜑 → (𝐷 ∈ 𝐴 ↔ (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
12	11	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → (𝐷 ∈ 𝐴 ↔ (𝒫 𝐷 ∩ Fin) ⊆ 𝐴))
13	5, 10, 12	3imtr4d 296	. . 3 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → (𝐶 ∈ 𝐴 → 𝐷 ∈ 𝐴))
14	13	impancom 454	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐷 ⊆ 𝐶 → 𝐷 ∈ 𝐴))
15	14	impr 457	1 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐶)) → 𝐷 ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1534   ∈ wcel 2113   ∖ cdif 3936   ∩ cin 3938   ⊆ wss 3939  𝒫 cpw 4542  ∪ cuni 4841  –1-1-onto→wf1o 6357  ‘cfv 6358  Fincfn 8512  cardccrd 9367

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-om 7584  df-1o 8105  df-en 8513  df-dom 8514  df-fin 8516

This theorem is referenced by:  ttukeylem6  9939  ttukeylem7  9940