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Theorem isfild 22559
 Description: Sufficient condition for a set of the form {𝑥 ∈ 𝒫 𝐴 ∣ 𝜑} 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 4500 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
32biimpri 231 . . . . . 6 (𝑥𝐴𝑥 ∈ 𝒫 𝐴)
43adantr 485 . . . . 5 ((𝑥𝐴𝜓) → 𝑥 ∈ 𝒫 𝐴)
51, 4syl6bi 256 . . . 4 (𝜑 → (𝑥𝐹𝑥 ∈ 𝒫 𝐴))
65ssrdv 3899 . . 3 (𝜑𝐹 ⊆ 𝒫 𝐴)
7 isfild.4 . . . 4 (𝜑 → ¬ [∅ / 𝑥]𝜓)
8 isfild.2 . . . . . 6 (𝜑𝐴𝑉)
91, 8isfildlem 22558 . . . . 5 (𝜑 → (∅ ∈ 𝐹 ↔ (∅ ⊆ 𝐴[∅ / 𝑥]𝜓)))
10 simpr 489 . . . . 5 ((∅ ⊆ 𝐴[∅ / 𝑥]𝜓) → [∅ / 𝑥]𝜓)
119, 10syl6bi 256 . . . 4 (𝜑 → (∅ ∈ 𝐹[∅ / 𝑥]𝜓))
127, 11mtod 201 . . 3 (𝜑 → ¬ ∅ ∈ 𝐹)
13 isfild.3 . . . . 5 (𝜑[𝐴 / 𝑥]𝜓)
14 ssid 3915 . . . . 5 𝐴𝐴
1513, 14jctil 524 . . . 4 (𝜑 → (𝐴𝐴[𝐴 / 𝑥]𝜓))
161, 8isfildlem 22558 . . . 4 (𝜑 → (𝐴𝐹 ↔ (𝐴𝐴[𝐴 / 𝑥]𝜓)))
1715, 16mpbird 260 . . 3 (𝜑𝐴𝐹)
186, 12, 173jca 1126 . 2 (𝜑 → (𝐹 ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈ 𝐹𝐴𝐹))
19 elpwi 4504 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
20 isfild.5 . . . . . . . . . . 11 ((𝜑𝑦𝐴𝑧𝑦) → ([𝑧 / 𝑥]𝜓[𝑦 / 𝑥]𝜓))
21 simp2 1135 . . . . . . . . . . 11 ((𝜑𝑦𝐴𝑧𝑦) → 𝑦𝐴)
2220, 21jctild 530 . . . . . . . . . 10 ((𝜑𝑦𝐴𝑧𝑦) → ([𝑧 / 𝑥]𝜓 → (𝑦𝐴[𝑦 / 𝑥]𝜓)))
2322adantld 495 . . . . . . . . 9 ((𝜑𝑦𝐴𝑧𝑦) → ((𝑧𝐴[𝑧 / 𝑥]𝜓) → (𝑦𝐴[𝑦 / 𝑥]𝜓)))
241, 8isfildlem 22558 . . . . . . . . . 10 (𝜑 → (𝑧𝐹 ↔ (𝑧𝐴[𝑧 / 𝑥]𝜓)))
25243ad2ant1 1131 . . . . . . . . 9 ((𝜑𝑦𝐴𝑧𝑦) → (𝑧𝐹 ↔ (𝑧𝐴[𝑧 / 𝑥]𝜓)))
261, 8isfildlem 22558 . . . . . . . . . 10 (𝜑 → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))
27263ad2ant1 1131 . . . . . . . . 9 ((𝜑𝑦𝐴𝑧𝑦) → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))
2823, 25, 273imtr4d 298 . . . . . . . 8 ((𝜑𝑦𝐴𝑧𝑦) → (𝑧𝐹𝑦𝐹))
29283expa 1116 . . . . . . 7 (((𝜑𝑦𝐴) ∧ 𝑧𝑦) → (𝑧𝐹𝑦𝐹))
3029impancom 456 . . . . . 6 (((𝜑𝑦𝐴) ∧ 𝑧𝐹) → (𝑧𝑦𝑦𝐹))
3130rexlimdva 3209 . . . . 5 ((𝜑𝑦𝐴) → (∃𝑧𝐹 𝑧𝑦𝑦𝐹))
3231ex 417 . . . 4 (𝜑 → (𝑦𝐴 → (∃𝑧𝐹 𝑧𝑦𝑦𝐹)))
3319, 32syl5 34 . . 3 (𝜑 → (𝑦 ∈ 𝒫 𝐴 → (∃𝑧𝐹 𝑧𝑦𝑦𝐹)))
3433ralrimiv 3113 . 2 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(∃𝑧𝐹 𝑧𝑦𝑦𝐹))
35 ssinss1 4143 . . . . . . 7 (𝑦𝐴 → (𝑦𝑧) ⊆ 𝐴)
3635ad2antrr 726 . . . . . 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 656 . . . . . 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 1118 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → (([𝑦 / 𝑥]𝜓[𝑧 / 𝑥]𝜓) → [(𝑦𝑧) / 𝑥]𝜓))
4140expimpd 458 . . . . . 6 (𝜑 → (((𝑦𝐴𝑧𝐴) ∧ ([𝑦 / 𝑥]𝜓[𝑧 / 𝑥]𝜓)) → [(𝑦𝑧) / 𝑥]𝜓))
4238, 41syl5bi 245 . . . . 5 (𝜑 → (((𝑦𝐴[𝑦 / 𝑥]𝜓) ∧ (𝑧𝐴[𝑧 / 𝑥]𝜓)) → [(𝑦𝑧) / 𝑥]𝜓))
4337, 42jcad 517 . . . 4 (𝜑 → (((𝑦𝐴[𝑦 / 𝑥]𝜓) ∧ (𝑧𝐴[𝑧 / 𝑥]𝜓)) → ((𝑦𝑧) ⊆ 𝐴[(𝑦𝑧) / 𝑥]𝜓)))
4426, 24anbi12d 634 . . . 4 (𝜑 → ((𝑦𝐹𝑧𝐹) ↔ ((𝑦𝐴[𝑦 / 𝑥]𝜓) ∧ (𝑧𝐴[𝑧 / 𝑥]𝜓))))
451, 8isfildlem 22558 . . . 4 (𝜑 → ((𝑦𝑧) ∈ 𝐹 ↔ ((𝑦𝑧) ⊆ 𝐴[(𝑦𝑧) / 𝑥]𝜓)))
4643, 44, 453imtr4d 298 . . 3 (𝜑 → ((𝑦𝐹𝑧𝐹) → (𝑦𝑧) ∈ 𝐹))
4746ralrimivv 3120 . 2 (𝜑 → ∀𝑦𝐹𝑧𝐹 (𝑦𝑧) ∈ 𝐹)
48 isfil2 22557 . 2 (𝐹 ∈ (Fil‘𝐴) ↔ ((𝐹 ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈ 𝐹𝐴𝐹) ∧ ∀𝑦 ∈ 𝒫 𝐴(∃𝑧𝐹 𝑧𝑦𝑦𝐹) ∧ ∀𝑦𝐹𝑧𝐹 (𝑦𝑧) ∈ 𝐹))
4918, 34, 47, 48syl3anbrc 1341 1 (𝜑𝐹 ∈ (Fil‘𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   ∈ wcel 2112  ∀wral 3071  ∃wrex 3072  [wsbc 3697   ∩ cin 3858   ⊆ wss 3859  ∅c0 4226  𝒫 cpw 4495  ‘cfv 6336  Filcfil 22546 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fv 6344  df-fbas 20164  df-fil 22547 This theorem is referenced by:  snfil  22565  fgcl  22579  filuni  22586  cfinfil  22594  csdfil  22595  supfil  22596  fin1aufil  22633
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