Theorem ttukeylem2 10423

Description: Lemma for ttukey 10431. 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 4555	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 ⊆ 𝐶) → 𝒫 𝐷 ⊆ 𝒫 𝐶)
3		ssrin 4183	. . . . 5 ⊢ (𝒫 𝐷 ⊆ 𝒫 𝐶 → (𝒫 𝐷 ∩ Fin) ⊆ (𝒫 𝐶 ∩ Fin))
4		sstr2 3929	. . . . 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 10422	. . . . 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 10422	. . . . 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 1540   ∈ wcel 2114   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851  –1-1-onto→wf1o 6491  ‘cfv 6492  Fincfn 8886  cardccrd 9850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7811  df-1o 8398  df-en 8887  df-dom 8888  df-fin 8890

This theorem is referenced by:  ttukeylem6  10427  ttukeylem7  10428