Theorem isfild 23854

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 → (𝑥 ∈ 𝐹 ↔ (𝑥 ⊆ 𝐴 ∧ 𝜓)))
isfild.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
isfild.3	⊢ (𝜑 → [𝐴 / 𝑥]𝜓)
isfild.4	⊢ (𝜑 → ¬ [∅ / 𝑥]𝜓)
isfild.5	⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → ([𝑧 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜓))
isfild.6	⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐴) → (([𝑦 / 𝑥]𝜓 ∧ [𝑧 / 𝑥]𝜓) → [(𝑦 ∩ 𝑧) / 𝑥]𝜓))

Assertion

Ref	Expression
isfild	⊢ (𝜑 → 𝐹 ∈ (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

Step	Hyp	Ref	Expression
1		isfild.1	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐹 ↔ (𝑥 ⊆ 𝐴 ∧ 𝜓)))
2		velpw 4613	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
3	2	biimpri 227	. . . . . 6 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝒫 𝐴)
4	3	adantr 479	. . . . 5 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝜓) → 𝑥 ∈ 𝒫 𝐴)
5	1, 4	biimtrdi 252	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐹 → 𝑥 ∈ 𝒫 𝐴))
6	5	ssrdv 3996	. . 3 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝐴)
7		isfild.4	. . . 4 ⊢ (𝜑 → ¬ [∅ / 𝑥]𝜓)
8		isfild.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
9	1, 8	isfildlem 23853	. . . . 5 ⊢ (𝜑 → (∅ ∈ 𝐹 ↔ (∅ ⊆ 𝐴 ∧ [∅ / 𝑥]𝜓)))
10		simpr 483	. . . . 5 ⊢ ((∅ ⊆ 𝐴 ∧ [∅ / 𝑥]𝜓) → [∅ / 𝑥]𝜓)
11	9, 10	biimtrdi 252	. . . 4 ⊢ (𝜑 → (∅ ∈ 𝐹 → [∅ / 𝑥]𝜓))
12	7, 11	mtod 197	. . 3 ⊢ (𝜑 → ¬ ∅ ∈ 𝐹)
13		isfild.3	. . . . 5 ⊢ (𝜑 → [𝐴 / 𝑥]𝜓)
14		ssid 4013	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
15	13, 14	jctil 518	. . . 4 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 ∧ [𝐴 / 𝑥]𝜓))
16	1, 8	isfildlem 23853	. . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝐹 ↔ (𝐴 ⊆ 𝐴 ∧ [𝐴 / 𝑥]𝜓)))
17	15, 16	mpbird 256	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐹)
18	6, 12, 17	3jca 1125	. 2 ⊢ (𝜑 → (𝐹 ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝐴 ∈ 𝐹))
19		elpwi 4615	. . . 4 ⊢ (𝑦 ∈ 𝒫 𝐴 → 𝑦 ⊆ 𝐴)
20		isfild.5	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → ([𝑧 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜓))
21		simp2 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → 𝑦 ⊆ 𝐴)
22	20, 21	jctild 524	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → ([𝑧 / 𝑥]𝜓 → (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)))
23	22	adantld 489	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → ((𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓) → (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)))
24	1, 8	isfildlem 23853	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ 𝐹 ↔ (𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓)))
25	24	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → (𝑧 ∈ 𝐹 ↔ (𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓)))
26	1, 8	isfildlem 23853	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝐹 ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)))
27	26	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → (𝑦 ∈ 𝐹 ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)))
28	23, 25, 27	3imtr4d 293	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → (𝑧 ∈ 𝐹 → 𝑦 ∈ 𝐹))
29	28	3expa 1115	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑧 ⊆ 𝑦) → (𝑧 ∈ 𝐹 → 𝑦 ∈ 𝐹))
30	29	impancom 450	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑧 ∈ 𝐹) → (𝑧 ⊆ 𝑦 → 𝑦 ∈ 𝐹))
31	30	rexlimdva 3148	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → (∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑦 → 𝑦 ∈ 𝐹))
32	31	ex 411	. . . 4 ⊢ (𝜑 → (𝑦 ⊆ 𝐴 → (∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑦 → 𝑦 ∈ 𝐹)))
33	19, 32	syl5 34	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝒫 𝐴 → (∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑦 → 𝑦 ∈ 𝐹)))
34	33	ralrimiv 3138	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑦 → 𝑦 ∈ 𝐹))
35		ssinss1 4247	. . . . . . 7 ⊢ (𝑦 ⊆ 𝐴 → (𝑦 ∩ 𝑧) ⊆ 𝐴)
36	35	ad2antrr 724	. . . . . 6 ⊢ (((𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓) ∧ (𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓)) → (𝑦 ∩ 𝑧) ⊆ 𝐴)
37	36	a1i 11	. . . . 5 ⊢ (𝜑 → (((𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓) ∧ (𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓)) → (𝑦 ∩ 𝑧) ⊆ 𝐴))
38		an4 654	. . . . . 6 ⊢ (((𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓) ∧ (𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓)) ↔ ((𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐴) ∧ ([𝑦 / 𝑥]𝜓 ∧ [𝑧 / 𝑥]𝜓)))
39		isfild.6	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐴) → (([𝑦 / 𝑥]𝜓 ∧ [𝑧 / 𝑥]𝜓) → [(𝑦 ∩ 𝑧) / 𝑥]𝜓))
40	39	3expb 1117	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐴)) → (([𝑦 / 𝑥]𝜓 ∧ [𝑧 / 𝑥]𝜓) → [(𝑦 ∩ 𝑧) / 𝑥]𝜓))
41	40	expimpd 452	. . . . . 6 ⊢ (𝜑 → (((𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐴) ∧ ([𝑦 / 𝑥]𝜓 ∧ [𝑧 / 𝑥]𝜓)) → [(𝑦 ∩ 𝑧) / 𝑥]𝜓))
42	38, 41	biimtrid 241	. . . . 5 ⊢ (𝜑 → (((𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓) ∧ (𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓)) → [(𝑦 ∩ 𝑧) / 𝑥]𝜓))
43	37, 42	jcad 511	. . . 4 ⊢ (𝜑 → (((𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓) ∧ (𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓)) → ((𝑦 ∩ 𝑧) ⊆ 𝐴 ∧ [(𝑦 ∩ 𝑧) / 𝑥]𝜓)))
44	26, 24	anbi12d 630	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹) ↔ ((𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓) ∧ (𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓))))
45	1, 8	isfildlem 23853	. . . 4 ⊢ (𝜑 → ((𝑦 ∩ 𝑧) ∈ 𝐹 ↔ ((𝑦 ∩ 𝑧) ⊆ 𝐴 ∧ [(𝑦 ∩ 𝑧) / 𝑥]𝜓)))
46	43, 44, 45	3imtr4d 293	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹) → (𝑦 ∩ 𝑧) ∈ 𝐹))
47	46	ralrimivv 3193	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 (𝑦 ∩ 𝑧) ∈ 𝐹)
48		isfil2 23852	. 2 ⊢ (𝐹 ∈ (Fil‘𝐴) ↔ ((𝐹 ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝐴 ∈ 𝐹) ∧ ∀𝑦 ∈ 𝒫 𝐴(∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑦 → 𝑦 ∈ 𝐹) ∧ ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 (𝑦 ∩ 𝑧) ∈ 𝐹))
49	18, 34, 47, 48	syl3anbrc 1340	1 ⊢ (𝜑 → 𝐹 ∈ (Fil‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   ∈ wcel 2100  ∀wral 3054  ∃wrex 3063  [wsbc 3787   ∩ cin 3957   ⊆ wss 3958  ∅c0 4333  𝒫 cpw 4608  ‘cfv 6557  Filcfil 23841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-sep 5305  ax-nul 5312  ax-pow 5371  ax-pr 5435

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-nel 3040  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-sbc 3788  df-csb 3904  df-dif 3961  df-un 3963  df-in 3965  df-ss 3975  df-nul 4334  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4917  df-br 5155  df-opab 5217  df-mpt 5238  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6509  df-fun 6559  df-fv 6565  df-fbas 21353  df-fil 23842

This theorem is referenced by:  snfil  23860  fgcl  23874  filuni  23881  cfinfil  23889  csdfil  23890  supfil  23891  fin1aufil  23928