MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isfild Structured version   Visualization version   GIF version

Theorem isfild 23887
Description: Sufficient condition for a set of the form {𝑥 ∈ 𝒫 𝐴𝜑} to be a filter. (Contributed by Mario Carneiro, 1-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.) (Revised by AV, 10-Apr-2024.)
Hypotheses
Ref Expression
isfild.1 (𝜑 → (𝑥𝐹 ↔ (𝑥𝐴𝜓)))
isfild.2 (𝜑𝐴𝑉)
isfild.3 (𝜑[𝐴 / 𝑥]𝜓)
isfild.4 (𝜑 → ¬ [∅ / 𝑥]𝜓)
isfild.5 ((𝜑𝑦𝐴𝑧𝑦) → ([𝑧 / 𝑥]𝜓[𝑦 / 𝑥]𝜓))
isfild.6 ((𝜑𝑦𝐴𝑧𝐴) → (([𝑦 / 𝑥]𝜓[𝑧 / 𝑥]𝜓) → [(𝑦𝑧) / 𝑥]𝜓))
Assertion
Ref Expression
isfild (𝜑𝐹 ∈ (Fil‘𝐴))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑧,𝐴   𝑥,𝐹,𝑦   𝑦,𝑧,𝐹   𝜑,𝑥,𝑦   𝜑,𝑧   𝜓,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem isfild
StepHypRef Expression
1 isfild.1 . . . . 5 (𝜑 → (𝑥𝐹 ↔ (𝑥𝐴𝜓)))
2 velpw 4627 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
32biimpri 228 . . . . . 6 (𝑥𝐴𝑥 ∈ 𝒫 𝐴)
43adantr 480 . . . . 5 ((𝑥𝐴𝜓) → 𝑥 ∈ 𝒫 𝐴)
51, 4biimtrdi 253 . . . 4 (𝜑 → (𝑥𝐹𝑥 ∈ 𝒫 𝐴))
65ssrdv 4014 . . 3 (𝜑𝐹 ⊆ 𝒫 𝐴)
7 isfild.4 . . . 4 (𝜑 → ¬ [∅ / 𝑥]𝜓)
8 isfild.2 . . . . . 6 (𝜑𝐴𝑉)
91, 8isfildlem 23886 . . . . 5 (𝜑 → (∅ ∈ 𝐹 ↔ (∅ ⊆ 𝐴[∅ / 𝑥]𝜓)))
10 simpr 484 . . . . 5 ((∅ ⊆ 𝐴[∅ / 𝑥]𝜓) → [∅ / 𝑥]𝜓)
119, 10biimtrdi 253 . . . 4 (𝜑 → (∅ ∈ 𝐹[∅ / 𝑥]𝜓))
127, 11mtod 198 . . 3 (𝜑 → ¬ ∅ ∈ 𝐹)
13 isfild.3 . . . . 5 (𝜑[𝐴 / 𝑥]𝜓)
14 ssid 4031 . . . . 5 𝐴𝐴
1513, 14jctil 519 . . . 4 (𝜑 → (𝐴𝐴[𝐴 / 𝑥]𝜓))
161, 8isfildlem 23886 . . . 4 (𝜑 → (𝐴𝐹 ↔ (𝐴𝐴[𝐴 / 𝑥]𝜓)))
1715, 16mpbird 257 . . 3 (𝜑𝐴𝐹)
186, 12, 173jca 1128 . 2 (𝜑 → (𝐹 ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈ 𝐹𝐴𝐹))
19 elpwi 4629 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
20 isfild.5 . . . . . . . . . . 11 ((𝜑𝑦𝐴𝑧𝑦) → ([𝑧 / 𝑥]𝜓[𝑦 / 𝑥]𝜓))
21 simp2 1137 . . . . . . . . . . 11 ((𝜑𝑦𝐴𝑧𝑦) → 𝑦𝐴)
2220, 21jctild 525 . . . . . . . . . 10 ((𝜑𝑦𝐴𝑧𝑦) → ([𝑧 / 𝑥]𝜓 → (𝑦𝐴[𝑦 / 𝑥]𝜓)))
2322adantld 490 . . . . . . . . 9 ((𝜑𝑦𝐴𝑧𝑦) → ((𝑧𝐴[𝑧 / 𝑥]𝜓) → (𝑦𝐴[𝑦 / 𝑥]𝜓)))
241, 8isfildlem 23886 . . . . . . . . . 10 (𝜑 → (𝑧𝐹 ↔ (𝑧𝐴[𝑧 / 𝑥]𝜓)))
25243ad2ant1 1133 . . . . . . . . 9 ((𝜑𝑦𝐴𝑧𝑦) → (𝑧𝐹 ↔ (𝑧𝐴[𝑧 / 𝑥]𝜓)))
261, 8isfildlem 23886 . . . . . . . . . 10 (𝜑 → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))
27263ad2ant1 1133 . . . . . . . . 9 ((𝜑𝑦𝐴𝑧𝑦) → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))
2823, 25, 273imtr4d 294 . . . . . . . 8 ((𝜑𝑦𝐴𝑧𝑦) → (𝑧𝐹𝑦𝐹))
29283expa 1118 . . . . . . 7 (((𝜑𝑦𝐴) ∧ 𝑧𝑦) → (𝑧𝐹𝑦𝐹))
3029impancom 451 . . . . . 6 (((𝜑𝑦𝐴) ∧ 𝑧𝐹) → (𝑧𝑦𝑦𝐹))
3130rexlimdva 3161 . . . . 5 ((𝜑𝑦𝐴) → (∃𝑧𝐹 𝑧𝑦𝑦𝐹))
3231ex 412 . . . 4 (𝜑 → (𝑦𝐴 → (∃𝑧𝐹 𝑧𝑦𝑦𝐹)))
3319, 32syl5 34 . . 3 (𝜑 → (𝑦 ∈ 𝒫 𝐴 → (∃𝑧𝐹 𝑧𝑦𝑦𝐹)))
3433ralrimiv 3151 . 2 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(∃𝑧𝐹 𝑧𝑦𝑦𝐹))
35 ssinss1 4267 . . . . . . 7 (𝑦𝐴 → (𝑦𝑧) ⊆ 𝐴)
3635ad2antrr 725 . . . . . 6 (((𝑦𝐴[𝑦 / 𝑥]𝜓) ∧ (𝑧𝐴[𝑧 / 𝑥]𝜓)) → (𝑦𝑧) ⊆ 𝐴)
3736a1i 11 . . . . 5 (𝜑 → (((𝑦𝐴[𝑦 / 𝑥]𝜓) ∧ (𝑧𝐴[𝑧 / 𝑥]𝜓)) → (𝑦𝑧) ⊆ 𝐴))
38 an4 655 . . . . . 6 (((𝑦𝐴[𝑦 / 𝑥]𝜓) ∧ (𝑧𝐴[𝑧 / 𝑥]𝜓)) ↔ ((𝑦𝐴𝑧𝐴) ∧ ([𝑦 / 𝑥]𝜓[𝑧 / 𝑥]𝜓)))
39 isfild.6 . . . . . . . 8 ((𝜑𝑦𝐴𝑧𝐴) → (([𝑦 / 𝑥]𝜓[𝑧 / 𝑥]𝜓) → [(𝑦𝑧) / 𝑥]𝜓))
40393expb 1120 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → (([𝑦 / 𝑥]𝜓[𝑧 / 𝑥]𝜓) → [(𝑦𝑧) / 𝑥]𝜓))
4140expimpd 453 . . . . . 6 (𝜑 → (((𝑦𝐴𝑧𝐴) ∧ ([𝑦 / 𝑥]𝜓[𝑧 / 𝑥]𝜓)) → [(𝑦𝑧) / 𝑥]𝜓))
4238, 41biimtrid 242 . . . . 5 (𝜑 → (((𝑦𝐴[𝑦 / 𝑥]𝜓) ∧ (𝑧𝐴[𝑧 / 𝑥]𝜓)) → [(𝑦𝑧) / 𝑥]𝜓))
4337, 42jcad 512 . . . 4 (𝜑 → (((𝑦𝐴[𝑦 / 𝑥]𝜓) ∧ (𝑧𝐴[𝑧 / 𝑥]𝜓)) → ((𝑦𝑧) ⊆ 𝐴[(𝑦𝑧) / 𝑥]𝜓)))
4426, 24anbi12d 631 . . . 4 (𝜑 → ((𝑦𝐹𝑧𝐹) ↔ ((𝑦𝐴[𝑦 / 𝑥]𝜓) ∧ (𝑧𝐴[𝑧 / 𝑥]𝜓))))
451, 8isfildlem 23886 . . . 4 (𝜑 → ((𝑦𝑧) ∈ 𝐹 ↔ ((𝑦𝑧) ⊆ 𝐴[(𝑦𝑧) / 𝑥]𝜓)))
4643, 44, 453imtr4d 294 . . 3 (𝜑 → ((𝑦𝐹𝑧𝐹) → (𝑦𝑧) ∈ 𝐹))
4746ralrimivv 3206 . 2 (𝜑 → ∀𝑦𝐹𝑧𝐹 (𝑦𝑧) ∈ 𝐹)
48 isfil2 23885 . 2 (𝐹 ∈ (Fil‘𝐴) ↔ ((𝐹 ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈ 𝐹𝐴𝐹) ∧ ∀𝑦 ∈ 𝒫 𝐴(∃𝑧𝐹 𝑧𝑦𝑦𝐹) ∧ ∀𝑦𝐹𝑧𝐹 (𝑦𝑧) ∈ 𝐹))
4918, 34, 47, 48syl3anbrc 1343 1 (𝜑𝐹 ∈ (Fil‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087  wcel 2108  wral 3067  wrex 3076  [wsbc 3804  cin 3975  wss 3976  c0 4352  𝒫 cpw 4622  cfv 6573  Filcfil 23874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fv 6581  df-fbas 21384  df-fil 23875
This theorem is referenced by:  snfil  23893  fgcl  23907  filuni  23914  cfinfil  23922  csdfil  23923  supfil  23924  fin1aufil  23961
  Copyright terms: Public domain W3C validator