Theorem ttukeylem1 10534

Description: Lemma for ttukey 10543. 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 3480	. . 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 4170	. . . . . . . 8 ⊢ ∪ 𝐴 ⊆ (∪ 𝐴 ∪ 𝐵)
5		undif1 4477	. . . . . . . 8 ⊢ ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) = (∪ 𝐴 ∪ 𝐵)
6	4, 5	sseqtrri 4014	. . . . . . 7 ⊢ ∪ 𝐴 ⊆ ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵)
7		fvex 6909	. . . . . . . . 9 ⊢ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ V
8		ttukeylem.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
9		f1ofo 6845	. . . . . . . . . 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		focdmex 7960	. . . . . . . . 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 7752	. . . . . . . 8 ⊢ (((∪ 𝐴 ∖ 𝐵) ∈ V ∧ 𝐵 ∈ 𝐴) → ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V)
15	12, 13, 14	syl2anc 582	. . . . . . 7 ⊢ (𝜑 → ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V)
16		ssexg 5324	. . . . . . 7 ⊢ ((∪ 𝐴 ⊆ ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) ∧ ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V) → ∪ 𝐴 ∈ V)
17	6, 15, 16	sylancr 585	. . . . . 6 ⊢ (𝜑 → ∪ 𝐴 ∈ V)
18		uniexb 7767	. . . . . 6 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
19	17, 18	sylibr 233	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
20		ssexg 5324	. . . . 5 ⊢ (((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 ∧ 𝐴 ∈ V) → (𝒫 𝐶 ∩ Fin) ∈ V)
21	3, 19, 20	syl2anr 595	. . . 4 ⊢ ((𝜑 ∧ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴) → (𝒫 𝐶 ∩ Fin) ∈ V)
22		infpwfidom 10053	. . . 4 ⊢ ((𝒫 𝐶 ∩ Fin) ∈ V → 𝐶 ≼ (𝒫 𝐶 ∩ Fin))
23		reldom 8970	. . . . 5 ⊢ Rel ≼
24	23	brrelex1i 5734	. . . 4 ⊢ (𝐶 ≼ (𝒫 𝐶 ∩ Fin) → 𝐶 ∈ V)
25	21, 22, 24	3syl 18	. . 3 ⊢ ((𝜑 ∧ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴) → 𝐶 ∈ V)
26	25	ex 411	. 2 ⊢ (𝜑 → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → 𝐶 ∈ V))
27		ttukeylem.3	. . 3 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
28		eleq1 2813	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
29		pweq 4618	. . . . . . 7 ⊢ (𝑥 = 𝐶 → 𝒫 𝑥 = 𝒫 𝐶)
30	29	ineq1d 4209	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝐶 ∩ Fin))
31	30	sseq1d 4008	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝒫 𝑥 ∩ Fin) ⊆ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
32	28, 31	bibi12d 344	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴) ↔ (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)))
33	32	spcgv 3580	. . 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 378	1 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394  ∀wal 1531   = wceq 1533   ∈ wcel 2098  Vcvv 3461   ∖ cdif 3941   ∪ cun 3942   ∩ cin 3943   ⊆ wss 3944  𝒫 cpw 4604  ∪ cuni 4909   class class class wbr 5149  –onto→wfo 6547  –1-1-onto→wf1o 6548  ‘cfv 6549   ≼ cdom 8962  Fincfn 8964  cardccrd 9960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-om 7872  df-1o 8487  df-en 8965  df-dom 8966  df-fin 8968

This theorem is referenced by:  ttukeylem2  10535  ttukeylem6  10539