Theorem ttukeylem1 10264

Description: Lemma for ttukey 10273. 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 3449	. . 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 4111	. . . . . . . 8 ⊢ ∪ 𝐴 ⊆ (∪ 𝐴 ∪ 𝐵)
5		undif1 4415	. . . . . . . 8 ⊢ ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) = (∪ 𝐴 ∪ 𝐵)
6	4, 5	sseqtrri 3963	. . . . . . 7 ⊢ ∪ 𝐴 ⊆ ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵)
7		fvex 6782	. . . . . . . . 9 ⊢ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ V
8		ttukeylem.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
9		f1ofo 6720	. . . . . . . . . 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 7790	. . . . . . . . 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 7591	. . . . . . . 8 ⊢ (((∪ 𝐴 ∖ 𝐵) ∈ V ∧ 𝐵 ∈ 𝐴) → ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V)
15	12, 13, 14	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V)
16		ssexg 5251	. . . . . . 7 ⊢ ((∪ 𝐴 ⊆ ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) ∧ ((∪ 𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V) → ∪ 𝐴 ∈ V)
17	6, 15, 16	sylancr 587	. . . . . 6 ⊢ (𝜑 → ∪ 𝐴 ∈ V)
18		uniexb 7606	. . . . . 6 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)
19	17, 18	sylibr 233	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
20		ssexg 5251	. . . . 5 ⊢ (((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 ∧ 𝐴 ∈ V) → (𝒫 𝐶 ∩ Fin) ∈ V)
21	3, 19, 20	syl2anr 597	. . . 4 ⊢ ((𝜑 ∧ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴) → (𝒫 𝐶 ∩ Fin) ∈ V)
22		infpwfidom 9783	. . . 4 ⊢ ((𝒫 𝐶 ∩ Fin) ∈ V → 𝐶 ≼ (𝒫 𝐶 ∩ Fin))
23		reldom 8720	. . . . 5 ⊢ Rel ≼
24	23	brrelex1i 5643	. . . 4 ⊢ (𝐶 ≼ (𝒫 𝐶 ∩ Fin) → 𝐶 ∈ V)
25	21, 22, 24	3syl 18	. . 3 ⊢ ((𝜑 ∧ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴) → 𝐶 ∈ V)
26	25	ex 413	. 2 ⊢ (𝜑 → ((𝒫 𝐶 ∩ Fin) ⊆ 𝐴 → 𝐶 ∈ V))
27		ttukeylem.3	. . 3 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
28		eleq1 2828	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
29		pweq 4555	. . . . . . 7 ⊢ (𝑥 = 𝐶 → 𝒫 𝑥 = 𝒫 𝐶)
30	29	ineq1d 4151	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝐶 ∩ Fin))
31	30	sseq1d 3957	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝒫 𝑥 ∩ Fin) ⊆ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
32	28, 31	bibi12d 346	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴) ↔ (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)))
33	32	spcgv 3534	. . 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 381	1 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1540   = wceq 1542   ∈ wcel 2110  Vcvv 3431   ∖ cdif 3889   ∪ cun 3890   ∩ cin 3891   ⊆ wss 3892  𝒫 cpw 4539  ∪ cuni 4845   class class class wbr 5079  –onto→wfo 6429  –1-1-onto→wf1o 6430  ‘cfv 6431   ≼ cdom 8712  Fincfn 8714  cardccrd 9692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6267  df-on 6268  df-lim 6269  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-om 7705  df-1o 8286  df-en 8715  df-dom 8716  df-fin 8718

This theorem is referenced by:  ttukeylem2  10265  ttukeylem6  10269