Theorem fival 8475

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		elex 3364	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		simpr 471	. . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 = ∩ 𝑥)
3		inss1 3982	. . . . . . . . . 10 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
4	3	sseli 3749	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ 𝒫 𝐴)
5	4	elpwid 4310	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ⊆ 𝐴)
6		eqvisset 3363	. . . . . . . . 9 ⊢ (𝑦 = ∩ 𝑥 → ∩ 𝑥 ∈ V)
7		intex 4952	. . . . . . . . 9 ⊢ (𝑥 ≠ ∅ ↔ ∩ 𝑥 ∈ V)
8	6, 7	sylibr 224	. . . . . . . 8 ⊢ (𝑦 = ∩ 𝑥 → 𝑥 ≠ ∅)
9		intssuni2 4637	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ⊆ ∪ 𝐴)
10	5, 8, 9	syl2an 577	. . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → ∩ 𝑥 ⊆ ∪ 𝐴)
11	2, 10	eqsstrd 3789	. . . . . 6 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 ⊆ ∪ 𝐴)
12		selpw 4305	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝐴)
13	11, 12	sylibr 224	. . . . 5 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 ∈ 𝒫 ∪ 𝐴)
14	13	rexlimiva 3176	. . . 4 ⊢ (∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥 → 𝑦 ∈ 𝒫 ∪ 𝐴)
15	14	abssi 3827	. . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥} ⊆ 𝒫 ∪ 𝐴
16		uniexg 7103	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
17		pwexg 4981	. . . 4 ⊢ (∪ 𝐴 ∈ V → 𝒫 ∪ 𝐴 ∈ V)
18	16, 17	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ V)
19		ssexg 4939	. . 3 ⊢ (({𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥} ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ V) → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥} ∈ V)
20	15, 18, 19	sylancr 569	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥} ∈ V)
21		pweq 4301	. . . . . 6 ⊢ (𝑧 = 𝐴 → 𝒫 𝑧 = 𝒫 𝐴)
22	21	ineq1d 3965	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝒫 𝑧 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
23	22	rexeqdv 3294	. . . 4 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = ∩ 𝑥 ↔ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥))
24	23	abbidv 2890	. . 3 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = ∩ 𝑥} = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥})
25		df-fi 8474	. . 3 ⊢ fi = (𝑧 ∈ V ↦ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = ∩ 𝑥})
26	24, 25	fvmptg 6423	. 2 ⊢ ((𝐴 ∈ V ∧ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥} ∈ V) → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥})
27	1, 20, 26	syl2anc 567	1 ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥})

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  {cab 2757   ≠ wne 2943  ∃wrex 3062  Vcvv 3351   ∩ cin 3723   ⊆ wss 3724  ∅c0 4064  𝒫 cpw 4298  ∪ cuni 4575  ∩ cint 4612  ‘cfv 6032  Fincfn 8110  ficfi 8473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-int 4613  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5995  df-fun 6034  df-fv 6040  df-fi 8474

This theorem is referenced by:  elfi  8476  fi0  8483