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Theorem isfild 21585
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.)
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 4142 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
32biimpri 218 . . . . . 6 (𝑥𝐴𝑥 ∈ 𝒫 𝐴)
43adantr 481 . . . . 5 ((𝑥𝐴𝜓) → 𝑥 ∈ 𝒫 𝐴)
51, 4syl6bi 243 . . . 4 (𝜑 → (𝑥𝐹𝑥 ∈ 𝒫 𝐴))
65ssrdv 3593 . . 3 (𝜑𝐹 ⊆ 𝒫 𝐴)
7 isfild.4 . . . 4 (𝜑 → ¬ [∅ / 𝑥]𝜓)
8 isfild.2 . . . . . 6 (𝜑𝐴 ∈ V)
91, 8isfildlem 21584 . . . . 5 (𝜑 → (∅ ∈ 𝐹 ↔ (∅ ⊆ 𝐴[∅ / 𝑥]𝜓)))
10 simpr 477 . . . . 5 ((∅ ⊆ 𝐴[∅ / 𝑥]𝜓) → [∅ / 𝑥]𝜓)
119, 10syl6bi 243 . . . 4 (𝜑 → (∅ ∈ 𝐹[∅ / 𝑥]𝜓))
127, 11mtod 189 . . 3 (𝜑 → ¬ ∅ ∈ 𝐹)
13 isfild.3 . . . . 5 (𝜑[𝐴 / 𝑥]𝜓)
14 ssid 3608 . . . . 5 𝐴𝐴
1513, 14jctil 559 . . . 4 (𝜑 → (𝐴𝐴[𝐴 / 𝑥]𝜓))
161, 8isfildlem 21584 . . . 4 (𝜑 → (𝐴𝐹 ↔ (𝐴𝐴[𝐴 / 𝑥]𝜓)))
1715, 16mpbird 247 . . 3 (𝜑𝐴𝐹)
186, 12, 173jca 1240 . 2 (𝜑 → (𝐹 ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈ 𝐹𝐴𝐹))
19 elpwi 4145 . . . 4 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
20 isfild.5 . . . . . . . . . . 11 ((𝜑𝑦𝐴𝑧𝑦) → ([𝑧 / 𝑥]𝜓[𝑦 / 𝑥]𝜓))
21 simp2 1060 . . . . . . . . . . 11 ((𝜑𝑦𝐴𝑧𝑦) → 𝑦𝐴)
2220, 21jctild 565 . . . . . . . . . 10 ((𝜑𝑦𝐴𝑧𝑦) → ([𝑧 / 𝑥]𝜓 → (𝑦𝐴[𝑦 / 𝑥]𝜓)))
2322adantld 483 . . . . . . . . 9 ((𝜑𝑦𝐴𝑧𝑦) → ((𝑧𝐴[𝑧 / 𝑥]𝜓) → (𝑦𝐴[𝑦 / 𝑥]𝜓)))
241, 8isfildlem 21584 . . . . . . . . . 10 (𝜑 → (𝑧𝐹 ↔ (𝑧𝐴[𝑧 / 𝑥]𝜓)))
25243ad2ant1 1080 . . . . . . . . 9 ((𝜑𝑦𝐴𝑧𝑦) → (𝑧𝐹 ↔ (𝑧𝐴[𝑧 / 𝑥]𝜓)))
261, 8isfildlem 21584 . . . . . . . . . 10 (𝜑 → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))
27263ad2ant1 1080 . . . . . . . . 9 ((𝜑𝑦𝐴𝑧𝑦) → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))
2823, 25, 273imtr4d 283 . . . . . . . 8 ((𝜑𝑦𝐴𝑧𝑦) → (𝑧𝐹𝑦𝐹))
29283expa 1262 . . . . . . 7 (((𝜑𝑦𝐴) ∧ 𝑧𝑦) → (𝑧𝐹𝑦𝐹))
3029impancom 456 . . . . . 6 (((𝜑𝑦𝐴) ∧ 𝑧𝐹) → (𝑧𝑦𝑦𝐹))
3130rexlimdva 3025 . . . . 5 ((𝜑𝑦𝐴) → (∃𝑧𝐹 𝑧𝑦𝑦𝐹))
3231ex 450 . . . 4 (𝜑 → (𝑦𝐴 → (∃𝑧𝐹 𝑧𝑦𝑦𝐹)))
3319, 32syl5 34 . . 3 (𝜑 → (𝑦 ∈ 𝒫 𝐴 → (∃𝑧𝐹 𝑧𝑦𝑦𝐹)))
3433ralrimiv 2960 . 2 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(∃𝑧𝐹 𝑧𝑦𝑦𝐹))
35 ssinss1 3824 . . . . . . 7 (𝑦𝐴 → (𝑦𝑧) ⊆ 𝐴)
3635ad2antrr 761 . . . . . 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 864 . . . . . 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 1263 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → (([𝑦 / 𝑥]𝜓[𝑧 / 𝑥]𝜓) → [(𝑦𝑧) / 𝑥]𝜓))
4140expimpd 628 . . . . . 6 (𝜑 → (((𝑦𝐴𝑧𝐴) ∧ ([𝑦 / 𝑥]𝜓[𝑧 / 𝑥]𝜓)) → [(𝑦𝑧) / 𝑥]𝜓))
4238, 41syl5bi 232 . . . . 5 (𝜑 → (((𝑦𝐴[𝑦 / 𝑥]𝜓) ∧ (𝑧𝐴[𝑧 / 𝑥]𝜓)) → [(𝑦𝑧) / 𝑥]𝜓))
4337, 42jcad 555 . . . 4 (𝜑 → (((𝑦𝐴[𝑦 / 𝑥]𝜓) ∧ (𝑧𝐴[𝑧 / 𝑥]𝜓)) → ((𝑦𝑧) ⊆ 𝐴[(𝑦𝑧) / 𝑥]𝜓)))
4426, 24anbi12d 746 . . . 4 (𝜑 → ((𝑦𝐹𝑧𝐹) ↔ ((𝑦𝐴[𝑦 / 𝑥]𝜓) ∧ (𝑧𝐴[𝑧 / 𝑥]𝜓))))
451, 8isfildlem 21584 . . . 4 (𝜑 → ((𝑦𝑧) ∈ 𝐹 ↔ ((𝑦𝑧) ⊆ 𝐴[(𝑦𝑧) / 𝑥]𝜓)))
4643, 44, 453imtr4d 283 . . 3 (𝜑 → ((𝑦𝐹𝑧𝐹) → (𝑦𝑧) ∈ 𝐹))
4746ralrimivv 2965 . 2 (𝜑 → ∀𝑦𝐹𝑧𝐹 (𝑦𝑧) ∈ 𝐹)
48 isfil2 21583 . 2 (𝐹 ∈ (Fil‘𝐴) ↔ ((𝐹 ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈ 𝐹𝐴𝐹) ∧ ∀𝑦 ∈ 𝒫 𝐴(∃𝑧𝐹 𝑧𝑦𝑦𝐹) ∧ ∀𝑦𝐹𝑧𝐹 (𝑦𝑧) ∈ 𝐹))
4918, 34, 47, 48syl3anbrc 1244 1 (𝜑𝐹 ∈ (Fil‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036  wcel 1987  wral 2907  wrex 2908  Vcvv 3189  [wsbc 3421  cin 3558  wss 3559  c0 3896  𝒫 cpw 4135  cfv 5852  Filcfil 21572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fv 5860  df-fbas 19675  df-fil 21573
This theorem is referenced by:  snfil  21591  fgcl  21605  filuni  21612  cfinfil  21620  csdfil  21621  supfil  21622  fin1aufil  21659
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