Theorem fival 7036

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 7035	. 2 ⊢ fi = (𝑧 ∈ V ↦ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = ∩ 𝑥})
2		pweq 3608	. . . . 5 ⊢ (𝑧 = 𝐴 → 𝒫 𝑧 = 𝒫 𝐴)
3	2	ineq1d 3363	. . . 4 ⊢ (𝑧 = 𝐴 → (𝒫 𝑧 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
4	3	rexeqdv 2700	. . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = ∩ 𝑥 ↔ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥))
5	4	abbidv 2314	. 2 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = ∩ 𝑥} = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥})
6		elex 2774	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
7		simpr 110	. . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 = ∩ 𝑥)
8		elinel1 3349	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ 𝒫 𝐴)
9	8	elpwid 3616	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ⊆ 𝐴)
10		eqvisset 2773	. . . . . . . . . . . 12 ⊢ (𝑦 = ∩ 𝑥 → ∩ 𝑥 ∈ V)
11		intexr 4183	. . . . . . . . . . . 12 ⊢ (∩ 𝑥 ∈ V → 𝑥 ≠ ∅)
12	10, 11	syl 14	. . . . . . . . . . 11 ⊢ (𝑦 = ∩ 𝑥 → 𝑥 ≠ ∅)
13	12	adantl 277	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑥 ≠ ∅)
14	13	neneqd 2388	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → ¬ 𝑥 = ∅)
15		elinel2 3350	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ Fin)
16	15	adantr 276	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑥 ∈ Fin)
17		fin0or 6947	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Fin → (𝑥 = ∅ ∨ ∃𝑧 𝑧 ∈ 𝑥))
18	17	orcomd 730	. . . . . . . . . 10 ⊢ (𝑥 ∈ Fin → (∃𝑧 𝑧 ∈ 𝑥 ∨ 𝑥 = ∅))
19	16, 18	syl 14	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → (∃𝑧 𝑧 ∈ 𝑥 ∨ 𝑥 = ∅))
20	14, 19	ecased 1360	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → ∃𝑧 𝑧 ∈ 𝑥)
21		intssuni2m 3898	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ ∃𝑧 𝑧 ∈ 𝑥) → ∩ 𝑥 ⊆ ∪ 𝐴)
22	9, 20, 21	syl2an2r 595	. . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → ∩ 𝑥 ⊆ ∪ 𝐴)
23	7, 22	eqsstrd 3219	. . . . . 6 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 ⊆ ∪ 𝐴)
24		velpw 3612	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝐴)
25	23, 24	sylibr 134	. . . . 5 ⊢ ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 ∈ 𝒫 ∪ 𝐴)
26	25	rexlimiva 2609	. . . 4 ⊢ (∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥 → 𝑦 ∈ 𝒫 ∪ 𝐴)
27	26	abssi 3258	. . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥} ⊆ 𝒫 ∪ 𝐴
28		uniexg 4474	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
29	28	pwexd 4214	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ V)
30		ssexg 4172	. . 3 ⊢ (({𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥} ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ V) → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥} ∈ V)
31	27, 29, 30	sylancr 414	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥} ∈ V)
32	1, 5, 6, 31	fvmptd3 5655	1 ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥})

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709   = wceq 1364  ∃wex 1506   ∈ wcel 2167  {cab 2182   ≠ wne 2367  ∃wrex 2476  Vcvv 2763   ∩ cin 3156   ⊆ wss 3157  ∅c0 3450  𝒫 cpw 3605  ∪ cuni 3839  ∩ cint 3874  ‘cfv 5258  Fincfn 6799  ficfi 7034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-er 6592  df-en 6800  df-fin 6802  df-fi 7035

This theorem is referenced by:  elfi  7037  fi0  7041