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Theorem isfild 21472
 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.)
Hypotheses
Ref Expression
isfild.1 (𝜑 → (𝑥𝐹 ↔ (𝑥𝐴𝜓)))
isfild.2 (𝜑𝐴 ∈ V)
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)

Proof of Theorem isfild
StepHypRef Expression
1 isfild.1 . . . . 5 (𝜑 → (𝑥𝐹 ↔ (𝑥𝐴𝜓)))
2 selpw 4115 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
32biimpri 217 . . . . . 6 (𝑥𝐴𝑥 ∈ 𝒫 𝐴)
43adantr 480 . . . . 5 ((𝑥𝐴𝜓) → 𝑥 ∈ 𝒫 𝐴)
51, 4syl6bi 242 . . . 4 (𝜑 → (𝑥𝐹𝑥 ∈ 𝒫 𝐴))
65ssrdv 3574 . . 3 (𝜑𝐹 ⊆ 𝒫 𝐴)
7 isfild.4 . . . 4 (𝜑 → ¬ [∅ / 𝑥]𝜓)
8 isfild.2 . . . . . 6 (𝜑𝐴 ∈ V)
91, 8isfildlem 21471 . . . . 5 (𝜑 → (∅ ∈ 𝐹 ↔ (∅ ⊆ 𝐴[∅ / 𝑥]𝜓)))
10 simpr 476 . . . . 5 ((∅ ⊆ 𝐴[∅ / 𝑥]𝜓) → [∅ / 𝑥]𝜓)
119, 10syl6bi 242 . . . 4 (𝜑 → (∅ ∈ 𝐹[∅ / 𝑥]𝜓))
127, 11mtod 188 . . 3 (𝜑 → ¬ ∅ ∈ 𝐹)
13 isfild.3 . . . . 5 (𝜑[𝐴 / 𝑥]𝜓)
14 ssid 3587 . . . . 5 𝐴𝐴
1513, 14jctil 558 . . . 4 (𝜑 → (𝐴𝐴[𝐴 / 𝑥]𝜓))
161, 8isfildlem 21471 . . . 4 (𝜑 → (𝐴𝐹 ↔ (𝐴𝐴[𝐴 / 𝑥]𝜓)))
1715, 16mpbird 246 . . 3 (𝜑𝐴𝐹)
186, 12, 173jca 1235 . 2 (𝜑 → (𝐹 ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈ 𝐹𝐴𝐹))
19 elpwi 4117 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
20 isfild.5 . . . . . . . . . . 11 ((𝜑𝑦𝐴𝑧𝑦) → ([𝑧 / 𝑥]𝜓[𝑦 / 𝑥]𝜓))
21 simp2 1055 . . . . . . . . . . 11 ((𝜑𝑦𝐴𝑧𝑦) → 𝑦𝐴)
2220, 21jctild 564 . . . . . . . . . 10 ((𝜑𝑦𝐴𝑧𝑦) → ([𝑧 / 𝑥]𝜓 → (𝑦𝐴[𝑦 / 𝑥]𝜓)))
2322adantld 482 . . . . . . . . 9 ((𝜑𝑦𝐴𝑧𝑦) → ((𝑧𝐴[𝑧 / 𝑥]𝜓) → (𝑦𝐴[𝑦 / 𝑥]𝜓)))
241, 8isfildlem 21471 . . . . . . . . . 10 (𝜑 → (𝑧𝐹 ↔ (𝑧𝐴[𝑧 / 𝑥]𝜓)))
25243ad2ant1 1075 . . . . . . . . 9 ((𝜑𝑦𝐴𝑧𝑦) → (𝑧𝐹 ↔ (𝑧𝐴[𝑧 / 𝑥]𝜓)))
261, 8isfildlem 21471 . . . . . . . . . 10 (𝜑 → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))
27263ad2ant1 1075 . . . . . . . . 9 ((𝜑𝑦𝐴𝑧𝑦) → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))
2823, 25, 273imtr4d 282 . . . . . . . 8 ((𝜑𝑦𝐴𝑧𝑦) → (𝑧𝐹𝑦𝐹))
29283expa 1257 . . . . . . 7 (((𝜑𝑦𝐴) ∧ 𝑧𝑦) → (𝑧𝐹𝑦𝐹))
3029impancom 455 . . . . . 6 (((𝜑𝑦𝐴) ∧ 𝑧𝐹) → (𝑧𝑦𝑦𝐹))
3130rexlimdva 3013 . . . . 5 ((𝜑𝑦𝐴) → (∃𝑧𝐹 𝑧𝑦𝑦𝐹))
3231ex 449 . . . 4 (𝜑 → (𝑦𝐴 → (∃𝑧𝐹 𝑧𝑦𝑦𝐹)))
3319, 32syl5 33 . . 3 (𝜑 → (𝑦 ∈ 𝒫 𝐴 → (∃𝑧𝐹 𝑧𝑦𝑦𝐹)))
3433ralrimiv 2948 . 2 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(∃𝑧𝐹 𝑧𝑦𝑦𝐹))
35 ssinss1 3803 . . . . . . 7 (𝑦𝐴 → (𝑦𝑧) ⊆ 𝐴)
3635ad2antrr 758 . . . . . 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 861 . . . . . 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 1258 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → (([𝑦 / 𝑥]𝜓[𝑧 / 𝑥]𝜓) → [(𝑦𝑧) / 𝑥]𝜓))
4140expimpd 627 . . . . . 6 (𝜑 → (((𝑦𝐴𝑧𝐴) ∧ ([𝑦 / 𝑥]𝜓[𝑧 / 𝑥]𝜓)) → [(𝑦𝑧) / 𝑥]𝜓))
4238, 41syl5bi 231 . . . . 5 (𝜑 → (((𝑦𝐴[𝑦 / 𝑥]𝜓) ∧ (𝑧𝐴[𝑧 / 𝑥]𝜓)) → [(𝑦𝑧) / 𝑥]𝜓))
4337, 42jcad 554 . . . 4 (𝜑 → (((𝑦𝐴[𝑦 / 𝑥]𝜓) ∧ (𝑧𝐴[𝑧 / 𝑥]𝜓)) → ((𝑦𝑧) ⊆ 𝐴[(𝑦𝑧) / 𝑥]𝜓)))
4426, 24anbi12d 743 . . . 4 (𝜑 → ((𝑦𝐹𝑧𝐹) ↔ ((𝑦𝐴[𝑦 / 𝑥]𝜓) ∧ (𝑧𝐴[𝑧 / 𝑥]𝜓))))
451, 8isfildlem 21471 . . . 4 (𝜑 → ((𝑦𝑧) ∈ 𝐹 ↔ ((𝑦𝑧) ⊆ 𝐴[(𝑦𝑧) / 𝑥]𝜓)))
4643, 44, 453imtr4d 282 . . 3 (𝜑 → ((𝑦𝐹𝑧𝐹) → (𝑦𝑧) ∈ 𝐹))
4746ralrimivv 2953 . 2 (𝜑 → ∀𝑦𝐹𝑧𝐹 (𝑦𝑧) ∈ 𝐹)
48 isfil2 21470 . 2 (𝐹 ∈ (Fil‘𝐴) ↔ ((𝐹 ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈ 𝐹𝐴𝐹) ∧ ∀𝑦 ∈ 𝒫 𝐴(∃𝑧𝐹 𝑧𝑦𝑦𝐹) ∧ ∀𝑦𝐹𝑧𝐹 (𝑦𝑧) ∈ 𝐹))
4918, 34, 47, 48syl3anbrc 1239 1 (𝜑𝐹 ∈ (Fil‘𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173  [wsbc 3402   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ‘cfv 5804  Filcfil 21459 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-fbas 19564  df-fil 21460 This theorem is referenced by:  snfil  21478  fgcl  21492  filuni  21499  cfinfil  21507  csdfil  21508  supfil  21509  fin1aufil  21546
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