Theorem fipwuni 9420

Description: The set of finite intersections of a set is contained in the powerset of the union of the elements of 𝐴. (Contributed by Mario Carneiro, 24-Nov-2013.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
fipwuni	⊢ (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴

Proof of Theorem fipwuni

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uniexg 7729	. . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
2	1	pwexd 5377	. . . 4 ⊢ (𝐴 ∈ V → 𝒫 ∪ 𝐴 ∈ V)
3		pwuni 4949	. . . 4 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
4		fiss 9418	. . . 4 ⊢ ((𝒫 ∪ 𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴) → (fi‘𝐴) ⊆ (fi‘𝒫 ∪ 𝐴))
5	2, 3, 4	sylancl 586	. . 3 ⊢ (𝐴 ∈ V → (fi‘𝐴) ⊆ (fi‘𝒫 ∪ 𝐴))
6		ssinss1 4237	. . . . . . 7 ⊢ (𝑥 ⊆ ∪ 𝐴 → (𝑥 ∩ 𝑦) ⊆ ∪ 𝐴)
7		vex 3478	. . . . . . . 8 ⊢ 𝑥 ∈ V
8	7	elpw 4606	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 ↔ 𝑥 ⊆ ∪ 𝐴)
9	7	inex1 5317	. . . . . . . 8 ⊢ (𝑥 ∩ 𝑦) ∈ V
10	9	elpw 4606	. . . . . . 7 ⊢ ((𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴 ↔ (𝑥 ∩ 𝑦) ⊆ ∪ 𝐴)
11	6, 8, 10	3imtr4i 291	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴)
12	11	adantr 481	. . . . 5 ⊢ ((𝑥 ∈ 𝒫 ∪ 𝐴 ∧ 𝑦 ∈ 𝒫 ∪ 𝐴) → (𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴)
13	12	rgen2 3197	. . . 4 ⊢ ∀𝑥 ∈ 𝒫 ∪ 𝐴∀𝑦 ∈ 𝒫 ∪ 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴
14		inficl 9419	. . . . 5 ⊢ (𝒫 ∪ 𝐴 ∈ V → (∀𝑥 ∈ 𝒫 ∪ 𝐴∀𝑦 ∈ 𝒫 ∪ 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴 ↔ (fi‘𝒫 ∪ 𝐴) = 𝒫 ∪ 𝐴))
15	2, 14	syl 17	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ 𝒫 ∪ 𝐴∀𝑦 ∈ 𝒫 ∪ 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴 ↔ (fi‘𝒫 ∪ 𝐴) = 𝒫 ∪ 𝐴))
16	13, 15	mpbii 232	. . 3 ⊢ (𝐴 ∈ V → (fi‘𝒫 ∪ 𝐴) = 𝒫 ∪ 𝐴)
17	5, 16	sseqtrd 4022	. 2 ⊢ (𝐴 ∈ V → (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴)
18		fvprc 6883	. . 3 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) = ∅)
19		0ss 4396	. . 3 ⊢ ∅ ⊆ 𝒫 ∪ 𝐴
20	18, 19	eqsstrdi 4036	. 2 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴)
21	17, 20	pm2.61i 182	1 ⊢ (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474   ∩ cin 3947   ⊆ wss 3948  ∅c0 4322  𝒫 cpw 4602  ∪ cuni 4908  ‘cfv 6543  ficfi 9404

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7855  df-1o 8465  df-er 8702  df-en 8939  df-fin 8942  df-fi 9405

This theorem is referenced by:  fiuni  9422  ordtbas  22695