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Theorem csdfil 23045
Description: The set of all elements whose complement is dominated by the base set is a filter. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
csdfil ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ≺ 𝑋} ∈ (Fil‘𝑋))
Distinct variable group:   𝑥,𝑋

Proof of Theorem csdfil
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difeq2 4051 . . . . . 6 (𝑥 = 𝑦 → (𝑋𝑥) = (𝑋𝑦))
21breq1d 5084 . . . . 5 (𝑥 = 𝑦 → ((𝑋𝑥) ≺ 𝑋 ↔ (𝑋𝑦) ≺ 𝑋))
32elrab 3624 . . . 4 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ≺ 𝑋} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝑋𝑦) ≺ 𝑋))
4 velpw 4538 . . . . 5 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
54anbi1i 624 . . . 4 ((𝑦 ∈ 𝒫 𝑋 ∧ (𝑋𝑦) ≺ 𝑋) ↔ (𝑦𝑋 ∧ (𝑋𝑦) ≺ 𝑋))
63, 5bitri 274 . . 3 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ≺ 𝑋} ↔ (𝑦𝑋 ∧ (𝑋𝑦) ≺ 𝑋))
76a1i 11 . 2 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ≺ 𝑋} ↔ (𝑦𝑋 ∧ (𝑋𝑦) ≺ 𝑋)))
8 simpl 483 . 2 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → 𝑋 ∈ dom card)
9 difid 4304 . . . 4 (𝑋𝑋) = ∅
10 infn0 9076 . . . . . 6 (ω ≼ 𝑋𝑋 ≠ ∅)
1110adantl 482 . . . . 5 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → 𝑋 ≠ ∅)
12 0sdomg 8891 . . . . . 6 (𝑋 ∈ dom card → (∅ ≺ 𝑋𝑋 ≠ ∅))
1312adantr 481 . . . . 5 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (∅ ≺ 𝑋𝑋 ≠ ∅))
1411, 13mpbird 256 . . . 4 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ∅ ≺ 𝑋)
159, 14eqbrtrid 5109 . . 3 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (𝑋𝑋) ≺ 𝑋)
16 difeq2 4051 . . . . . 6 (𝑦 = 𝑋 → (𝑋𝑦) = (𝑋𝑋))
1716breq1d 5084 . . . . 5 (𝑦 = 𝑋 → ((𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑋) ≺ 𝑋))
1817sbcieg 3756 . . . 4 (𝑋 ∈ dom card → ([𝑋 / 𝑦](𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑋) ≺ 𝑋))
1918adantr 481 . . 3 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ([𝑋 / 𝑦](𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑋) ≺ 𝑋))
2015, 19mpbird 256 . 2 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → [𝑋 / 𝑦](𝑋𝑦) ≺ 𝑋)
21 sdomirr 8901 . . 3 ¬ 𝑋𝑋
22 0ex 5231 . . . . 5 ∅ ∈ V
23 difeq2 4051 . . . . . . 7 (𝑦 = ∅ → (𝑋𝑦) = (𝑋 ∖ ∅))
24 dif0 4306 . . . . . . 7 (𝑋 ∖ ∅) = 𝑋
2523, 24eqtrdi 2794 . . . . . 6 (𝑦 = ∅ → (𝑋𝑦) = 𝑋)
2625breq1d 5084 . . . . 5 (𝑦 = ∅ → ((𝑋𝑦) ≺ 𝑋𝑋𝑋))
2722, 26sbcie 3759 . . . 4 ([∅ / 𝑦](𝑋𝑦) ≺ 𝑋𝑋𝑋)
2827a1i 11 . . 3 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ([∅ / 𝑦](𝑋𝑦) ≺ 𝑋𝑋𝑋))
2921, 28mtbiri 327 . 2 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ¬ [∅ / 𝑦](𝑋𝑦) ≺ 𝑋)
30 simp1l 1196 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → 𝑋 ∈ dom card)
3130difexd 5253 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → (𝑋𝑤) ∈ V)
32 sscon 4073 . . . . . 6 (𝑤𝑧 → (𝑋𝑧) ⊆ (𝑋𝑤))
33323ad2ant3 1134 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → (𝑋𝑧) ⊆ (𝑋𝑤))
34 ssdomg 8786 . . . . 5 ((𝑋𝑤) ∈ V → ((𝑋𝑧) ⊆ (𝑋𝑤) → (𝑋𝑧) ≼ (𝑋𝑤)))
3531, 33, 34sylc 65 . . . 4 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → (𝑋𝑧) ≼ (𝑋𝑤))
36 domsdomtr 8899 . . . . 5 (((𝑋𝑧) ≼ (𝑋𝑤) ∧ (𝑋𝑤) ≺ 𝑋) → (𝑋𝑧) ≺ 𝑋)
3736ex 413 . . . 4 ((𝑋𝑧) ≼ (𝑋𝑤) → ((𝑋𝑤) ≺ 𝑋 → (𝑋𝑧) ≺ 𝑋))
3835, 37syl 17 . . 3 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → ((𝑋𝑤) ≺ 𝑋 → (𝑋𝑧) ≺ 𝑋))
39 vex 3436 . . . 4 𝑤 ∈ V
40 difeq2 4051 . . . . 5 (𝑦 = 𝑤 → (𝑋𝑦) = (𝑋𝑤))
4140breq1d 5084 . . . 4 (𝑦 = 𝑤 → ((𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑤) ≺ 𝑋))
4239, 41sbcie 3759 . . 3 ([𝑤 / 𝑦](𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑤) ≺ 𝑋)
43 vex 3436 . . . 4 𝑧 ∈ V
44 difeq2 4051 . . . . 5 (𝑦 = 𝑧 → (𝑋𝑦) = (𝑋𝑧))
4544breq1d 5084 . . . 4 (𝑦 = 𝑧 → ((𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑧) ≺ 𝑋))
4643, 45sbcie 3759 . . 3 ([𝑧 / 𝑦](𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑧) ≺ 𝑋)
4738, 42, 463imtr4g 296 . 2 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → ([𝑤 / 𝑦](𝑋𝑦) ≺ 𝑋[𝑧 / 𝑦](𝑋𝑦) ≺ 𝑋))
48 infunsdom 9970 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ ((𝑋𝑧) ≺ 𝑋 ∧ (𝑋𝑤) ≺ 𝑋)) → ((𝑋𝑧) ∪ (𝑋𝑤)) ≺ 𝑋)
4948ex 413 . . . . 5 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (((𝑋𝑧) ≺ 𝑋 ∧ (𝑋𝑤) ≺ 𝑋) → ((𝑋𝑧) ∪ (𝑋𝑤)) ≺ 𝑋))
50 difindi 4215 . . . . . 6 (𝑋 ∖ (𝑧𝑤)) = ((𝑋𝑧) ∪ (𝑋𝑤))
5150breq1i 5081 . . . . 5 ((𝑋 ∖ (𝑧𝑤)) ≺ 𝑋 ↔ ((𝑋𝑧) ∪ (𝑋𝑤)) ≺ 𝑋)
5249, 51syl6ibr 251 . . . 4 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (((𝑋𝑧) ≺ 𝑋 ∧ (𝑋𝑤) ≺ 𝑋) → (𝑋 ∖ (𝑧𝑤)) ≺ 𝑋))
53523ad2ant1 1132 . . 3 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑋) → (((𝑋𝑧) ≺ 𝑋 ∧ (𝑋𝑤) ≺ 𝑋) → (𝑋 ∖ (𝑧𝑤)) ≺ 𝑋))
5446, 42anbi12i 627 . . 3 (([𝑧 / 𝑦](𝑋𝑦) ≺ 𝑋[𝑤 / 𝑦](𝑋𝑦) ≺ 𝑋) ↔ ((𝑋𝑧) ≺ 𝑋 ∧ (𝑋𝑤) ≺ 𝑋))
5543inex1 5241 . . . 4 (𝑧𝑤) ∈ V
56 difeq2 4051 . . . . 5 (𝑦 = (𝑧𝑤) → (𝑋𝑦) = (𝑋 ∖ (𝑧𝑤)))
5756breq1d 5084 . . . 4 (𝑦 = (𝑧𝑤) → ((𝑋𝑦) ≺ 𝑋 ↔ (𝑋 ∖ (𝑧𝑤)) ≺ 𝑋))
5855, 57sbcie 3759 . . 3 ([(𝑧𝑤) / 𝑦](𝑋𝑦) ≺ 𝑋 ↔ (𝑋 ∖ (𝑧𝑤)) ≺ 𝑋)
5953, 54, 583imtr4g 296 . 2 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑋) → (([𝑧 / 𝑦](𝑋𝑦) ≺ 𝑋[𝑤 / 𝑦](𝑋𝑦) ≺ 𝑋) → [(𝑧𝑤) / 𝑦](𝑋𝑦) ≺ 𝑋))
607, 8, 20, 29, 47, 59isfild 23009 1 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ≺ 𝑋} ∈ (Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wne 2943  {crab 3068  Vcvv 3432  [wsbc 3716  cdif 3884  cun 3885  cin 3886  wss 3887  c0 4256  𝒫 cpw 4533   class class class wbr 5074  dom cdm 5589  cfv 6433  ωcom 7712  cdom 8731  csdm 8732  cardccrd 9693  Filcfil 22996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-oi 9269  df-dju 9659  df-card 9697  df-fbas 20594  df-fil 22997
This theorem is referenced by:  ufilen  23081
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