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Theorem supfil 23843
Description: The supersets of a nonempty set which are also subsets of a given base set form a filter. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
Assertion
Ref Expression
supfil ((𝐴𝑉𝐵𝐴𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴𝐵𝑥} ∈ (Fil‘𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem supfil
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq2 4003 . . . . 5 (𝑥 = 𝑦 → (𝐵𝑥𝐵𝑦))
21elrab 3679 . . . 4 (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴𝐵𝑥} ↔ (𝑦 ∈ 𝒫 𝐴𝐵𝑦))
3 velpw 4609 . . . . 5 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
43anbi1i 622 . . . 4 ((𝑦 ∈ 𝒫 𝐴𝐵𝑦) ↔ (𝑦𝐴𝐵𝑦))
52, 4bitri 274 . . 3 (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴𝐵𝑥} ↔ (𝑦𝐴𝐵𝑦))
65a1i 11 . 2 ((𝐴𝑉𝐵𝐴𝐵 ≠ ∅) → (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴𝐵𝑥} ↔ (𝑦𝐴𝐵𝑦)))
7 simp1 1133 . 2 ((𝐴𝑉𝐵𝐴𝐵 ≠ ∅) → 𝐴𝑉)
8 simp2 1134 . . 3 ((𝐴𝑉𝐵𝐴𝐵 ≠ ∅) → 𝐵𝐴)
9 sseq2 4003 . . . . 5 (𝑦 = 𝐴 → (𝐵𝑦𝐵𝐴))
109sbcieg 3814 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑦]𝐵𝑦𝐵𝐴))
117, 10syl 17 . . 3 ((𝐴𝑉𝐵𝐴𝐵 ≠ ∅) → ([𝐴 / 𝑦]𝐵𝑦𝐵𝐴))
128, 11mpbird 256 . 2 ((𝐴𝑉𝐵𝐴𝐵 ≠ ∅) → [𝐴 / 𝑦]𝐵𝑦)
13 ss0 4400 . . . . 5 (𝐵 ⊆ ∅ → 𝐵 = ∅)
1413necon3ai 2954 . . . 4 (𝐵 ≠ ∅ → ¬ 𝐵 ⊆ ∅)
15143ad2ant3 1132 . . 3 ((𝐴𝑉𝐵𝐴𝐵 ≠ ∅) → ¬ 𝐵 ⊆ ∅)
16 0ex 5308 . . . 4 ∅ ∈ V
17 sseq2 4003 . . . 4 (𝑦 = ∅ → (𝐵𝑦𝐵 ⊆ ∅))
1816, 17sbcie 3817 . . 3 ([∅ / 𝑦]𝐵𝑦𝐵 ⊆ ∅)
1915, 18sylnibr 328 . 2 ((𝐴𝑉𝐵𝐴𝐵 ≠ ∅) → ¬ [∅ / 𝑦]𝐵𝑦)
20 sstr 3985 . . . . 5 ((𝐵𝑤𝑤𝑧) → 𝐵𝑧)
2120expcom 412 . . . 4 (𝑤𝑧 → (𝐵𝑤𝐵𝑧))
22 vex 3465 . . . . 5 𝑤 ∈ V
23 sseq2 4003 . . . . 5 (𝑦 = 𝑤 → (𝐵𝑦𝐵𝑤))
2422, 23sbcie 3817 . . . 4 ([𝑤 / 𝑦]𝐵𝑦𝐵𝑤)
25 vex 3465 . . . . 5 𝑧 ∈ V
26 sseq2 4003 . . . . 5 (𝑦 = 𝑧 → (𝐵𝑦𝐵𝑧))
2725, 26sbcie 3817 . . . 4 ([𝑧 / 𝑦]𝐵𝑦𝐵𝑧)
2821, 24, 273imtr4g 295 . . 3 (𝑤𝑧 → ([𝑤 / 𝑦]𝐵𝑦[𝑧 / 𝑦]𝐵𝑦))
29283ad2ant3 1132 . 2 (((𝐴𝑉𝐵𝐴𝐵 ≠ ∅) ∧ 𝑧𝐴𝑤𝑧) → ([𝑤 / 𝑦]𝐵𝑦[𝑧 / 𝑦]𝐵𝑦))
30 ssin 4229 . . . . . 6 ((𝐵𝑧𝐵𝑤) ↔ 𝐵 ⊆ (𝑧𝑤))
3130biimpi 215 . . . . 5 ((𝐵𝑧𝐵𝑤) → 𝐵 ⊆ (𝑧𝑤))
3227, 24, 31syl2anb 596 . . . 4 (([𝑧 / 𝑦]𝐵𝑦[𝑤 / 𝑦]𝐵𝑦) → 𝐵 ⊆ (𝑧𝑤))
3325inex1 5318 . . . . 5 (𝑧𝑤) ∈ V
34 sseq2 4003 . . . . 5 (𝑦 = (𝑧𝑤) → (𝐵𝑦𝐵 ⊆ (𝑧𝑤)))
3533, 34sbcie 3817 . . . 4 ([(𝑧𝑤) / 𝑦]𝐵𝑦𝐵 ⊆ (𝑧𝑤))
3632, 35sylibr 233 . . 3 (([𝑧 / 𝑦]𝐵𝑦[𝑤 / 𝑦]𝐵𝑦) → [(𝑧𝑤) / 𝑦]𝐵𝑦)
3736a1i 11 . 2 (((𝐴𝑉𝐵𝐴𝐵 ≠ ∅) ∧ 𝑧𝐴𝑤𝐴) → (([𝑧 / 𝑦]𝐵𝑦[𝑤 / 𝑦]𝐵𝑦) → [(𝑧𝑤) / 𝑦]𝐵𝑦))
386, 7, 12, 19, 29, 37isfild 23806 1 ((𝐴𝑉𝐵𝐴𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴𝐵𝑥} ∈ (Fil‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084  wcel 2098  wne 2929  {crab 3418  [wsbc 3773  cin 3943  wss 3944  c0 4322  𝒫 cpw 4604  cfv 6549  Filcfil 23793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fv 6557  df-fbas 21293  df-fil 23794
This theorem is referenced by:  fclscf  23973  flimfnfcls  23976
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