Theorem isfild 22150

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 → (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)

Proof of Theorem isfild

Step	Hyp	Ref	Expression
1		isfild.1	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐹 ↔ (𝑥 ⊆ 𝐴 ∧ 𝜓)))
2		selpw 4460	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
3	2	biimpri 229	. . . . . 6 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝒫 𝐴)
4	3	adantr 481	. . . . 5 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝜓) → 𝑥 ∈ 𝒫 𝐴)
5	1, 4	syl6bi 254	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐹 → 𝑥 ∈ 𝒫 𝐴))
6	5	ssrdv 3895	. . 3 ⊢ (𝜑 → 𝐹 ⊆ 𝒫 𝐴)
7		isfild.4	. . . 4 ⊢ (𝜑 → ¬ [∅ / 𝑥]𝜓)
8		isfild.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
9	1, 8	isfildlem 22149	. . . . 5 ⊢ (𝜑 → (∅ ∈ 𝐹 ↔ (∅ ⊆ 𝐴 ∧ [∅ / 𝑥]𝜓)))
10		simpr 485	. . . . 5 ⊢ ((∅ ⊆ 𝐴 ∧ [∅ / 𝑥]𝜓) → [∅ / 𝑥]𝜓)
11	9, 10	syl6bi 254	. . . 4 ⊢ (𝜑 → (∅ ∈ 𝐹 → [∅ / 𝑥]𝜓))
12	7, 11	mtod 199	. . 3 ⊢ (𝜑 → ¬ ∅ ∈ 𝐹)
13		isfild.3	. . . . 5 ⊢ (𝜑 → [𝐴 / 𝑥]𝜓)
14		ssid 3910	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
15	13, 14	jctil 520	. . . 4 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 ∧ [𝐴 / 𝑥]𝜓))
16	1, 8	isfildlem 22149	. . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝐹 ↔ (𝐴 ⊆ 𝐴 ∧ [𝐴 / 𝑥]𝜓)))
17	15, 16	mpbird 258	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐹)
18	6, 12, 17	3jca 1121	. 2 ⊢ (𝜑 → (𝐹 ⊆ 𝒫 𝐴 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝐴 ∈ 𝐹))
19		elpwi 4463	. . . 4 ⊢ (𝑦 ∈ 𝒫 𝐴 → 𝑦 ⊆ 𝐴)
20		isfild.5	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → ([𝑧 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜓))
21		simp2 1130	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → 𝑦 ⊆ 𝐴)
22	20, 21	jctild 526	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → ([𝑧 / 𝑥]𝜓 → (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)))
23	22	adantld 491	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → ((𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓) → (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)))
24	1, 8	isfildlem 22149	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ 𝐹 ↔ (𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓)))
25	24	3ad2ant1 1126	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → (𝑧 ∈ 𝐹 ↔ (𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓)))
26	1, 8	isfildlem 22149	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝐹 ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)))
27	26	3ad2ant1 1126	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → (𝑦 ∈ 𝐹 ↔ (𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓)))
28	23, 25, 27	3imtr4d 295	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → (𝑧 ∈ 𝐹 → 𝑦 ∈ 𝐹))
29	28	3expa 1111	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑧 ⊆ 𝑦) → (𝑧 ∈ 𝐹 → 𝑦 ∈ 𝐹))
30	29	impancom 452	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑧 ∈ 𝐹) → (𝑧 ⊆ 𝑦 → 𝑦 ∈ 𝐹))
31	30	rexlimdva 3247	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → (∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑦 → 𝑦 ∈ 𝐹))
32	31	ex 413	. . . 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 3148	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑦 → 𝑦 ∈ 𝐹))
35		ssinss1 4134	. . . . . . 7 ⊢ (𝑦 ⊆ 𝐴 → (𝑦 ∩ 𝑧) ⊆ 𝐴)
36	35	ad2antrr 722	. . . . . 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 652	. . . . . 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 1113	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐴)) → (([𝑦 / 𝑥]𝜓 ∧ [𝑧 / 𝑥]𝜓) → [(𝑦 ∩ 𝑧) / 𝑥]𝜓))
41	40	expimpd 454	. . . . . 6 ⊢ (𝜑 → (((𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐴) ∧ ([𝑦 / 𝑥]𝜓 ∧ [𝑧 / 𝑥]𝜓)) → [(𝑦 ∩ 𝑧) / 𝑥]𝜓))
42	38, 41	syl5bi 243	. . . . 5 ⊢ (𝜑 → (((𝑦 ⊆ 𝐴 ∧ [𝑦 / 𝑥]𝜓) ∧ (𝑧 ⊆ 𝐴 ∧ [𝑧 / 𝑥]𝜓)) → [(𝑦 ∩ 𝑧) / 𝑥]𝜓))
43	37, 42	jcad 513	. . . 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 22149	. . . 4 ⊢ (𝜑 → ((𝑦 ∩ 𝑧) ∈ 𝐹 ↔ ((𝑦 ∩ 𝑧) ⊆ 𝐴 ∧ [(𝑦 ∩ 𝑧) / 𝑥]𝜓)))
46	43, 44, 45	3imtr4d 295	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹) → (𝑦 ∩ 𝑧) ∈ 𝐹))
47	46	ralrimivv 3157	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 (𝑦 ∩ 𝑧) ∈ 𝐹)
48		isfil2 22148	. 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 1336	1 ⊢ (𝜑 → 𝐹 ∈ (Fil‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   ∈ wcel 2081  ∀wral 3105  ∃wrex 3106  Vcvv 3437  [wsbc 3706   ∩ cin 3858   ⊆ wss 3859  ∅c0 4211  𝒫 cpw 4453  ‘cfv 6225  Filcfil 22137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fv 6233  df-fbas 20224  df-fil 22138

This theorem is referenced by:  snfil  22156  fgcl  22170  filuni  22177  cfinfil  22185  csdfil  22186  supfil  22187  fin1aufil  22224