Theorem isfild 24084

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 4562	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
3	2	biranri 511	. . . . 5 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝜓) → 𝑥 ∈ 𝒫 𝐴)
4	1, 3	biimtrdi 256	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐹 → 𝑥 ∈ 𝒫 𝐴))
5	4	ssrdv 3937	. . 3 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝐴)
6		isfild.4	. . . 4 ⊢ (𝜑 → ¬ [∅ / 𝑥]𝜓)
7		isfild.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8	1, 7	isfildlem 24083	. . . . 5 ⊢ (𝜑 → (∅ ∈ 𝐹 ↔ (∅ ⊆ 𝐴 ∧ [∅ / 𝑥]𝜓)))
9		simpr 490	. . . . 5 ⊢ ((∅ ⊆ 𝐴 ∧ [∅ / 𝑥]𝜓) → [∅ / 𝑥]𝜓)
10	8, 9	biimtrdi 256	. . . 4 ⊢ (𝜑 → (∅ ∈ 𝐹 → [∅ / 𝑥]𝜓))
11	6, 10	mtod 201	. . 3 ⊢ (𝜑 → ¬ ∅ ∈ 𝐹)
12		isfild.3	. . . . 5 ⊢ (𝜑 → [𝐴 / 𝑥]𝜓)
13		ssid 3953	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
14	12, 13	jctil 529	. . . 4 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 ∧ [𝐴 / 𝑥]𝜓))
15	1, 7	isfildlem 24083	. . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝐹 ↔ (𝐴 ⊆ 𝐴 ∧ [𝐴 / 𝑥]𝜓)))
16	14, 15	mpbird 260	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐹)
17	5, 11, 16	3jca 1146	. 2 ⊢ (𝜑 → (𝐹 ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝐴 ∈ 𝐹))
18		elpwi 4564	. . . 4 ⊢ (𝑦 ∈ 𝒫 𝐴 → 𝑦 ⊆ 𝐴)
19		isfild.5	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → ([𝑧 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜓))
20		simp2 1155	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → 𝑦 ⊆ 𝐴)
21	19, 20	jctild 535	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → ([𝑧 / 𝑥]𝜓 → (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)))
22	21	adantld 496	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → ((𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓) → (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)))
23	1, 7	isfildlem 24083	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ 𝐹 ↔ (𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓)))
24	23	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → (𝑧 ∈ 𝐹 ↔ (𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓)))
25	1, 7	isfildlem 24083	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝐹 ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)))
26	25	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → (𝑦 ∈ 𝐹 ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)))
27	22, 24, 26	3imtr4d 297	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → (𝑧 ∈ 𝐹 → 𝑦 ∈ 𝐹))
28	27	3expa 1136	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑧 ⊆ 𝑦) → (𝑧 ∈ 𝐹 → 𝑦 ∈ 𝐹))
29	28	impancom 457	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑧 ∈ 𝐹) → (𝑧 ⊆ 𝑦 → 𝑦 ∈ 𝐹))
30	29	rexlimdva 3163	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → (∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑦 → 𝑦 ∈ 𝐹))
31	30	ex 418	. . . 4 ⊢ (𝜑 → (𝑦 ⊆ 𝐴 → (∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑦 → 𝑦 ∈ 𝐹)))
32	18, 31	syl5 35	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝒫 𝐴 → (∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑦 → 𝑦 ∈ 𝐹)))
33	32	ralrimiv 3153	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑦 → 𝑦 ∈ 𝐹))
34		ssinss1 4191	. . . . . . 7 ⊢ (𝑦 ⊆ 𝐴 → (𝑦 ∩ 𝑧) ⊆ 𝐴)
35	34	ad2antrr 739	. . . . . 6 ⊢ (((𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓) ∧ (𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓)) → (𝑦 ∩ 𝑧) ⊆ 𝐴)
36	35	a1i 11	. . . . 5 ⊢ (𝜑 → (((𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓) ∧ (𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓)) → (𝑦 ∩ 𝑧) ⊆ 𝐴))
37		an4 669	. . . . . 6 ⊢ (((𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓) ∧ (𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓)) ↔ ((𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐴) ∧ ([𝑦 / 𝑥]𝜓 ∧ [𝑧 / 𝑥]𝜓)))
38		isfild.6	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐴) → (([𝑦 / 𝑥]𝜓 ∧ [𝑧 / 𝑥]𝜓) → [(𝑦 ∩ 𝑧) / 𝑥]𝜓))
39	38	3expb 1138	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐴)) → (([𝑦 / 𝑥]𝜓 ∧ [𝑧 / 𝑥]𝜓) → [(𝑦 ∩ 𝑧) / 𝑥]𝜓))
40	39	expimpd 459	. . . . . 6 ⊢ (𝜑 → (((𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐴) ∧ ([𝑦 / 𝑥]𝜓 ∧ [𝑧 / 𝑥]𝜓)) → [(𝑦 ∩ 𝑧) / 𝑥]𝜓))
41	37, 40	biimtrid 245	. . . . 5 ⊢ (𝜑 → (((𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓) ∧ (𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓)) → [(𝑦 ∩ 𝑧) / 𝑥]𝜓))
42	36, 41	jcad 522	. . . 4 ⊢ (𝜑 → (((𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓) ∧ (𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓)) → ((𝑦 ∩ 𝑧) ⊆ 𝐴 ∧ [(𝑦 ∩ 𝑧) / 𝑥]𝜓)))
43	25, 23	anbi12d 644	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹) ↔ ((𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓) ∧ (𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓))))
44	1, 7	isfildlem 24083	. . . 4 ⊢ (𝜑 → ((𝑦 ∩ 𝑧) ∈ 𝐹 ↔ ((𝑦 ∩ 𝑧) ⊆ 𝐴 ∧ [(𝑦 ∩ 𝑧) / 𝑥]𝜓)))
45	42, 43, 44	3imtr4d 297	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹) → (𝑦 ∩ 𝑧) ∈ 𝐹))
46	45	ralrimivv 3203	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 (𝑦 ∩ 𝑧) ∈ 𝐹)
47		isfil2 24082	. 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))
48	17, 33, 46, 47	syl3anbrc 1362	1 ⊢ (𝜑 → 𝐹 ∈ (Fil‘𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  [wsbc 3739   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  ‘cfv 6533  Filcfil 24071

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fv 6541  df-fbas 21582  df-fil 24072

This theorem is used by:  snfil  24090  fgcl  24104  filuni  24111  cfinfil  24119  csdfil  24120  supfil  24121  fin1aufil  24158