Theorem fipwuni 9317

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 7679	. . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
2	1	pwexd 5319	. . . 4 ⊢ (𝐴 ∈ V → 𝒫 ∪ 𝐴 ∈ V)
3		pwuni 4896	. . . 4 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
4		fiss 9315	. . . 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 4195	. . . . . . 7 ⊢ (𝑥 ⊆ ∪ 𝐴 → (𝑥 ∩ 𝑦) ⊆ ∪ 𝐴)
7		vex 3441	. . . . . . . 8 ⊢ 𝑥 ∈ V
8	7	elpw 4553	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 ↔ 𝑥 ⊆ ∪ 𝐴)
9	7	inex1 5257	. . . . . . . 8 ⊢ (𝑥 ∩ 𝑦) ∈ V
10	9	elpw 4553	. . . . . . 7 ⊢ ((𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴 ↔ (𝑥 ∩ 𝑦) ⊆ ∪ 𝐴)
11	6, 8, 10	3imtr4i 292	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴)
12	11	adantr 480	. . . . 5 ⊢ ((𝑥 ∈ 𝒫 ∪ 𝐴 ∧ 𝑦 ∈ 𝒫 ∪ 𝐴) → (𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴)
13	12	rgen2 3173	. . . 4 ⊢ ∀𝑥 ∈ 𝒫 ∪ 𝐴∀𝑦 ∈ 𝒫 ∪ 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 ∪ 𝐴
14		inficl 9316	. . . . 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 233	. . 3 ⊢ (𝐴 ∈ V → (fi‘𝒫 ∪ 𝐴) = 𝒫 ∪ 𝐴)
17	5, 16	sseqtrd 3967	. 2 ⊢ (𝐴 ∈ V → (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴)
18		fvprc 6820	. . 3 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) = ∅)
19		0ss 4349	. . 3 ⊢ ∅ ⊆ 𝒫 ∪ 𝐴
20	18, 19	eqsstrdi 3975	. 2 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴)
21	17, 20	pm2.61i 182	1 ⊢ (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1541   ∈ wcel 2113  ∀wral 3048  Vcvv 3437   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282  𝒫 cpw 4549  ∪ cuni 4858  ‘cfv 6486  ficfi 9301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-om 7803  df-1o 8391  df-2o 8392  df-en 8876  df-fin 8879  df-fi 9302

This theorem is referenced by:  fiuni  9319  ordtbas  23108