Theorem filint2 10446

Description: A filter is closed under taking finite intersections.

Assertion

Ref	Expression
filint2	⊢ (F ∈ Fil → ((A ⊆ F ⋀ A ≠ ∅ ⋀ ∃x ∈ ω A ≈ x) → ∩A ∈ F))

Distinct variable groups:   x,A   x,F

Proof of Theorem filint2

Step	Hyp	Ref	Expression
1		elpw2g 2717	. . . . . . 7 ⊢ (F ∈ Fil → (A ∈ ℘F ↔ A ⊆ F))
2		sseq1 2072	. . . . . . . . 9 ⊢ (y = A → (y ⊆ F ↔ A ⊆ F))
3		neeq1 1582	. . . . . . . . . . 11 ⊢ (y = A → (y ≠ ∅ ↔ A ≠ ∅))
4		breq1 2612	. . . . . . . . . . . . 13 ⊢ (y = A → (y ≈ x ↔ A ≈ x))
5	4	rexbidv 1656	. . . . . . . . . . . 12 ⊢ (y = A → (∃x ∈ ω y ≈ x ↔ ∃x ∈ ω A ≈ x))
6		inteq 2526	. . . . . . . . . . . . 13 ⊢ (y = A → ∩y = ∩A)
7	6	eleq1d 1532	. . . . . . . . . . . 12 ⊢ (y = A → (∩y ∈ F ↔ ∩A ∈ F))
8	5, 7	imbi12d 624	. . . . . . . . . . 11 ⊢ (y = A → ((∃x ∈ ω y ≈ x → ∩y ∈ F) ↔ (∃x ∈ ω A ≈ x → ∩A ∈ F)))
9	3, 8	imbi12d 624	. . . . . . . . . 10 ⊢ (y = A → ((y ≠ ∅ → (∃x ∈ ω y ≈ x → ∩y ∈ F)) ↔ (A ≠ ∅ → (∃x ∈ ω A ≈ x → ∩A ∈ F))))
10	9	imbi2d 610	. . . . . . . . 9 ⊢ (y = A → ((F ∈ Fil → (y ≠ ∅ → (∃x ∈ ω y ≈ x → ∩y ∈ F))) ↔ (F ∈ Fil → (A ≠ ∅ → (∃x ∈ ω A ≈ x → ∩A ∈ F)))))
11	2, 10	imbi12d 624	. . . . . . . 8 ⊢ (y = A → ((y ⊆ F → (F ∈ Fil → (y ≠ ∅ → (∃x ∈ ω y ≈ x → ∩y ∈ F)))) ↔ (A ⊆ F → (F ∈ Fil → (A ≠ ∅ → (∃x ∈ ω A ≈ x → ∩A ∈ F))))))
12		eqid 1468	. . . . . . . . . . . . 13 ⊢ ∪F = ∪F
13	12	isfil 10433	. . . . . . . . . . . 12 ⊢ (F ∈ Fil → (F ∈ Fil ↔ ((¬ ∅ ∈ F ⋀ ∪F ∈ F) ⋀ ∀y∀x((y ∈ F ⋀ x ⊆ ∪F ⋀ y ⊆ x) → x ∈ F) ⋀ ∀y ∈ F ∀x ∈ F (y ∩ x) ∈ F)))
14		3simp3 788	. . . . . . . . . . . . . 14 ⊢ (((¬ ∅ ∈ F ⋀ ∪F ∈ F) ⋀ ∀y∀x((y ∈ F ⋀ x ⊆ ∪F ⋀ y ⊆ x) → x ∈ F) ⋀ ∀y ∈ F ∀x ∈ F (y ∩ x) ∈ F) → ∀y ∈ F ∀x ∈ F (y ∩ x) ∈ F)
15		fiint 4534	. . . . . . . . . . . . . 14 ⊢ (∀y ∈ F ∀x ∈ F (y ∩ x) ∈ F ↔ ∀y((y ⊆ F ⋀ y ≠ ∅ ⋀ ∃x ∈ ω y ≈ x) → ∩y ∈ F))
16	14, 15	sylib 198	. . . . . . . . . . . . 13 ⊢ (((¬ ∅ ∈ F ⋀ ∪F ∈ F) ⋀ ∀y∀x((y ∈ F ⋀ x ⊆ ∪F ⋀ y ⊆ x) → x ∈ F) ⋀ ∀y ∈ F ∀x ∈ F (y ∩ x) ∈ F) → ∀y((y ⊆ F ⋀ y ≠ ∅ ⋀ ∃x ∈ ω y ≈ x) → ∩y ∈ F))
17	16	19.21bi 1056	. . . . . . . . . . . 12 ⊢ (((¬ ∅ ∈ F ⋀ ∪F ∈ F) ⋀ ∀y∀x((y ∈ F ⋀ x ⊆ ∪F ⋀ y ⊆ x) → x ∈ F) ⋀ ∀y ∈ F ∀x ∈ F (y ∩ x) ∈ F) → ((y ⊆ F ⋀ y ≠ ∅ ⋀ ∃x ∈ ω y ≈ x) → ∩y ∈ F))
18	13, 17	syl6bi 214	. . . . . . . . . . 11 ⊢ (F ∈ Fil → (F ∈ Fil → ((y ⊆ F ⋀ y ≠ ∅ ⋀ ∃x ∈ ω y ≈ x) → ∩y ∈ F)))
19	18	pm2.43i 64	. . . . . . . . . 10 ⊢ (F ∈ Fil → ((y ⊆ F ⋀ y ≠ ∅ ⋀ ∃x ∈ ω y ≈ x) → ∩y ∈ F))
20	19	3expd 848	. . . . . . . . 9 ⊢ (F ∈ Fil → (y ⊆ F → (y ≠ ∅ → (∃x ∈ ω y ≈ x → ∩y ∈ F))))
21	20	com12 11	. . . . . . . 8 ⊢ (y ⊆ F → (F ∈ Fil → (y ≠ ∅ → (∃x ∈ ω y ≈ x → ∩y ∈ F))))
22	11, 21	vtoclg 1838	. . . . . . 7 ⊢ (A ∈ ℘F → (A ⊆ F → (F ∈ Fil → (A ≠ ∅ → (∃x ∈ ω A ≈ x → ∩A ∈ F)))))
23	1, 22	syl6bir 215	. . . . . 6 ⊢ (F ∈ Fil → (A ⊆ F → (A ⊆ F → (F ∈ Fil → (A ≠ ∅ → (∃x ∈ ω A ≈ x → ∩A ∈ F))))))
24	23	com4l 39	. . . . 5 ⊢ (A ⊆ F → (A ⊆ F → (F ∈ Fil → (F ∈ Fil → (A ≠ ∅ → (∃x ∈ ω A ≈ x → ∩A ∈ F))))))
25	24	pm2.43i 64	. . . 4 ⊢ (A ⊆ F → (F ∈ Fil → (F ∈ Fil → (A ≠ ∅ → (∃x ∈ ω A ≈ x → ∩A ∈ F)))))
26	25	com13 33	. . 3 ⊢ (F ∈ Fil → (F ∈ Fil → (A ⊆ F → (A ≠ ∅ → (∃x ∈ ω A ≈ x → ∩A ∈ F)))))
27	26	pm2.43i 64	. 2 ⊢ (F ∈ Fil → (A ⊆ F → (A ≠ ∅ → (∃x ∈ ω A ≈ x → ∩A ∈ F))))
28	27	3impd 845	1 ⊢ (F ∈ Fil → ((A ⊆ F ⋀ A ≠ ∅ ⋀ ∃x ∈ ω A ≈ x) → ∩A ∈ F))

Syntax hints:  ¬ wn 2   → wi 3   ⋀ wa 223   ⋀ w3a 773  ∀wal 951   = wceq 953   ∈ wcel 955   ≠ wne 1577  ∀wral 1637  ∃wrex 1638   ∩ cin 2036   ⊆ wss 2037  ∅c0 2270  ℘cpw 2391  ∪cuni 2493  ∩cint 2523   class class class wbr 2609  ωcom 3121   ≈ cen 4348  Filcfil 10431

This theorem is referenced by:  cnfilca 10451

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1o 4117  df-oadd 4119  df-er 4245  df-en 4351  df-fil 10432

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