Theorem ttukeylem1 10196

Description: Lemma for ttukey 10205. Expand out the property of being an element of a property of finite character. (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
ttukeylem1	⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ttukeylem1

Step	Hyp	Ref	Expression
1		elex 3440	. . 3 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V)
2	1	a1i 11	. 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ V))
3		id 22	. . . . 5 ⊢ ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)
4		ssun1 4102	. . . . . . . 8 ⊢ ∪ 𝐴 ⊆ (∪ 𝐴 ∪ 𝐵)
5		undif1 4406	. . . . . . . 8 ⊢ ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) = (∪ 𝐴 ∪ 𝐵)
6	4, 5	sseqtrri 3954	. . . . . . 7 ⊢ ∪ 𝐴 ⊆ ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵)
7		fvex 6769	. . . . . . . . 9 ⊢ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ V
8		ttukeylem.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
9		f1ofo 6707	. . . . . . . . . 10 ⊢ (𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵) → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–onto→(∪ 𝐴 ∖ 𝐵))
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–onto→(∪ 𝐴 ∖ 𝐵))
11		fornex 7772	. . . . . . . . 9 ⊢ ((card‘(∪ 𝐴 ∖ 𝐵)) ∈ V → (𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–onto→(∪ 𝐴 ∖ 𝐵) → (∪ 𝐴 ∖ 𝐵) ∈ V))
12	7, 10, 11	mpsyl 68	. . . . . . . 8 ⊢ (𝜑 → (∪ 𝐴 ∖ 𝐵) ∈ V)
13		ttukeylem.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
14		unexg 7577	. . . . . . . 8 ⊢ (((∪ 𝐴 ∖ 𝐵) ∈ V ∧ 𝐵 ∈ 𝐴) → ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V)
15	12, 13, 14	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V)
16		ssexg 5242	. . . . . . 7 ⊢ ((∪ 𝐴 ⊆ ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) ∧ ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V) → ∪ 𝐴 ∈ V)
17	6, 15, 16	sylancr 586	. . . . . 6 ⊢ (𝜑 → ∪ 𝐴 ∈ V)
18		uniexb 7592	. . . . . 6 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
19	17, 18	sylibr 233	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
20		ssexg 5242	. . . . 5 ⊢ (((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 ∧ 𝐴 ∈ V) → (𝒫 𝐶 ∩ Fin) ∈ V)
21	3, 19, 20	syl2anr 596	. . . 4 ⊢ ((𝜑 ∧ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴) → (𝒫 𝐶 ∩ Fin) ∈ V)
22		infpwfidom 9715	. . . 4 ⊢ ((𝒫 𝐶 ∩ Fin) ∈ V → 𝐶 ≼ (𝒫 𝐶 ∩ Fin))
23		reldom 8697	. . . . 5 ⊢ Rel ≼
24	23	brrelex1i 5634	. . . 4 ⊢ (𝐶 ≼ (𝒫 𝐶 ∩ Fin) → 𝐶 ∈ V)
25	21, 22, 24	3syl 18	. . 3 ⊢ ((𝜑 ∧ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴) → 𝐶 ∈ V)
26	25	ex 412	. 2 ⊢ (𝜑 → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → 𝐶 ∈ V))
27		ttukeylem.3	. . 3 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
28		eleq1 2826	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
29		pweq 4546	. . . . . . 7 ⊢ (𝑥 = 𝐶 → 𝒫 𝑥 = 𝒫 𝐶)
30	29	ineq1d 4142	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝐶 ∩ Fin))
31	30	sseq1d 3948	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝒫 𝑥 ∩ Fin) ⊆ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
32	28, 31	bibi12d 345	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴) ↔ (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)))
33	32	spcgv 3525	. . 3 ⊢ (𝐶 ∈ V → (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴) → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)))
34	27, 33	syl5com 31	. 2 ⊢ (𝜑 → (𝐶 ∈ V → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)))
35	2, 26, 34	pm5.21ndd 380	1 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836   class class class wbr 5070  –onto→wfo 6416  –1-1-onto→wf1o 6417  ‘cfv 6418   ≼ cdom 8689  Fincfn 8691  cardccrd 9624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267  df-en 8692  df-dom 8693  df-fin 8695

This theorem is referenced by:  ttukeylem2  10197  ttukeylem6  10201