Theorem ttukeylem1 10106

Description: Lemma for ttukey 10115. 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 3419	. . 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 4076	. . . . . . . 8 ⊢ ∪ 𝐴 ⊆ (∪ 𝐴 ∪ 𝐵)
5		undif1 4380	. . . . . . . 8 ⊢ ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) = (∪ 𝐴 ∪ 𝐵)
6	4, 5	sseqtrri 3928	. . . . . . 7 ⊢ ∪ 𝐴 ⊆ ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵)
7		fvex 6719	. . . . . . . . 9 ⊢ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ V
8		ttukeylem.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
9		f1ofo 6657	. . . . . . . . . 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 7718	. . . . . . . . 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 7523	. . . . . . . 8 ⊢ (((∪ 𝐴 ∖ 𝐵) ∈ V ∧ 𝐵 ∈ 𝐴) → ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V)
15	12, 13, 14	syl2anc 587	. . . . . . 7 ⊢ (𝜑 → ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V)
16		ssexg 5205	. . . . . . 7 ⊢ ((∪ 𝐴 ⊆ ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) ∧ ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V) → ∪ 𝐴 ∈ V)
17	6, 15, 16	sylancr 590	. . . . . 6 ⊢ (𝜑 → ∪ 𝐴 ∈ V)
18		uniexb 7538	. . . . . 6 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
19	17, 18	sylibr 237	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
20		ssexg 5205	. . . . 5 ⊢ (((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 ∧ 𝐴 ∈ V) → (𝒫 𝐶 ∩ Fin) ∈ V)
21	3, 19, 20	syl2anr 600	. . . 4 ⊢ ((𝜑 ∧ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴) → (𝒫 𝐶 ∩ Fin) ∈ V)
22		infpwfidom 9625	. . . 4 ⊢ ((𝒫 𝐶 ∩ Fin) ∈ V → 𝐶 ≼ (𝒫 𝐶 ∩ Fin))
23		reldom 8621	. . . . 5 ⊢ Rel ≼
24	23	brrelex1i 5594	. . . 4 ⊢ (𝐶 ≼ (𝒫 𝐶 ∩ Fin) → 𝐶 ∈ V)
25	21, 22, 24	3syl 18	. . 3 ⊢ ((𝜑 ∧ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴) → 𝐶 ∈ V)
26	25	ex 416	. 2 ⊢ (𝜑 → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → 𝐶 ∈ V))
27		ttukeylem.3	. . 3 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
28		eleq1 2821	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
29		pweq 4519	. . . . . . 7 ⊢ (𝑥 = 𝐶 → 𝒫 𝑥 = 𝒫 𝐶)
30	29	ineq1d 4116	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝐶 ∩ Fin))
31	30	sseq1d 3922	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝒫 𝑥 ∩ Fin) ⊆ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
32	28, 31	bibi12d 349	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴) ↔ (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)))
33	32	spcgv 3504	. . 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 384	1 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541   = wceq 1543   ∈ wcel 2110  Vcvv 3401   ∖ cdif 3854   ∪ cun 3855   ∩ cin 3856   ⊆ wss 3857  𝒫 cpw 4503  ∪ cuni 4809   class class class wbr 5043  –onto→wfo 6367  –1-1-onto→wf1o 6368  ‘cfv 6369   ≼ cdom 8613  Fincfn 8615  cardccrd 9534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-om 7634  df-1o 8191  df-en 8616  df-dom 8617  df-fin 8619

This theorem is referenced by:  ttukeylem2  10107  ttukeylem6  10111