Theorem fival 9358

Description: The set of all the finite intersections of the elements of 𝐴. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)

Assertion

Ref	Expression
fival	⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥})

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑉

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem fival

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fi 9357	. 2 ⊢ fi = (𝑧 ∈ V ↦ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = ∩ 𝑥})
2		pweq 4569	. . . . 5 ⊢ (𝑧 = 𝐴 → 𝒫 𝑧 = 𝒫 𝐴)
3	2	ineq1d 4171	. . . 4 ⊢ (𝑧 = 𝐴 → (𝒫 𝑧 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
4	3	rexeqdv 3321	. . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = ∩ 𝑥 ↔ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥))
5	4	abbidv 2828	. 2 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = ∩ 𝑥} = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥})
6		elex 3475	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
7		simpr 488	. . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 = ∩ 𝑥)
8		elinel1 4153	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ 𝒫 𝐴)
9	8	elpwid 4564	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ⊆ 𝐴)
10		eqvisset 3474	. . . . . . . . 9 ⊢ (𝑦 = ∩ 𝑥 → ∩ 𝑥 ∈ V)
11		intex 5300	. . . . . . . . 9 ⊢ (𝑥 ≠ ∅ ↔ ∩ 𝑥 ∈ V)
12	10, 11	sylibr 236	. . . . . . . 8 ⊢ (𝑦 = ∩ 𝑥 → 𝑥 ≠ ∅)
13		intssuni2 4931	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ⊆ ∪ 𝐴)
14	9, 12, 13	syl2an 605	. . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → ∩ 𝑥 ⊆ ∪ 𝐴)
15	7, 14	eqsstrd 3970	. . . . . 6 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 ⊆ ∪ 𝐴)
16		velpw 4560	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝐴)
17	15, 16	sylibr 236	. . . . 5 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 ∈ 𝒫 ∪ 𝐴)
18	17	rexlimiva 3155	. . . 4 ⊢ (∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥 → 𝑦 ∈ 𝒫 ∪ 𝐴)
19	18	abssi 4021	. . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥} ⊆ 𝒫 ∪ 𝐴
20		uniexg 7723	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
21	20	pwexd 5336	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ V)
22		ssexg 5279	. . 3 ⊢ (({𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥} ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ V) → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥} ∈ V)
23	19, 21, 22	sylancr 596	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥} ∈ V)
24	1, 5, 6, 23	fvmptd3 6999	1 ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  {cab 2740   ≠ wne 2957  ∃wrex 3086  Vcvv 3454   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4555  ∪ cuni 4865  ∩ cint 4905  ‘cfv 6521  Fincfn 8927  ficfi 9356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-iota 6477  df-fun 6523  df-fv 6529  df-fi 9357

This theorem is referenced by:  elfi  9359  fi0  9366