Theorem fmcfil 23070

Description: The Cauchy filter condition for a filter map. (Contributed by Mario Carneiro, 13-Oct-2015.)

Assertion

Ref	Expression
fmcfil	⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (((𝑋 FilMap 𝐹)‘𝐵) ∈ (CauFil‘𝐷) ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥))

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐵   𝑤,𝐹,𝑥,𝑦,𝑧   𝑤,𝑋,𝑥,𝑦,𝑧   𝑤,𝑌,𝑥,𝑦,𝑧   𝑤,𝐷,𝑥,𝑦,𝑧

Proof of Theorem fmcfil

Dummy variables 𝑢 𝑠 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6220	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
2		fmval 21747	. . . 4 ⊢ ((𝑋 ∈ dom ∞Met ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))))
3	1, 2	syl3an1 1359	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))))
4	3	eleq1d 2686	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (((𝑋 FilMap 𝐹)‘𝐵) ∈ (CauFil‘𝐷) ↔ (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))) ∈ (CauFil‘𝐷)))
5		simp1 1061	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐷 ∈ (∞Met‘𝑋))
6		simp2 1062	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐵 ∈ (fBas‘𝑌))
7		simp3 1063	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐹:𝑌⟶𝑋)
8	1	3ad2ant1 1082	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝑋 ∈ dom ∞Met)
9		eqid 2622	. . . . 5 ⊢ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) = ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))
10	9	fbasrn 21688	. . . 4 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋 ∧ 𝑋 ∈ dom ∞Met) → ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋))
11	6, 7, 8, 10	syl3anc 1326	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋))
12		fgcfil 23069	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋)) → ((𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))) ∈ (CauFil‘𝐷) ↔ ∀𝑥 ∈ ℝ+ ∃𝑠 ∈ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 (𝑢𝐷𝑣) < 𝑥))
13	5, 11, 12	syl2anc 693	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))) ∈ (CauFil‘𝐷) ↔ ∀𝑥 ∈ ℝ+ ∃𝑠 ∈ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 (𝑢𝐷𝑣) < 𝑥))
14		imassrn 5477	. . . . . . . 8 ⊢ (𝐹 “ 𝑦) ⊆ ran 𝐹
15		frn 6053	. . . . . . . . 9 ⊢ (𝐹:𝑌⟶𝑋 → ran 𝐹 ⊆ 𝑋)
16	15	3ad2ant3 1084	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ran 𝐹 ⊆ 𝑋)
17	14, 16	syl5ss 3614	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐹 “ 𝑦) ⊆ 𝑋)
18	8, 17	ssexd 4805	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐹 “ 𝑦) ∈ V)
19	18	ralrimivw 2967	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ∀𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ∈ V)
20		eqid 2622	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) = (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))
21		raleq 3138	. . . . . . 7 ⊢ (𝑠 = (𝐹 “ 𝑦) → (∀𝑣 ∈ 𝑠 (𝑢𝐷𝑣) < 𝑥 ↔ ∀𝑣 ∈ (𝐹 “ 𝑦)(𝑢𝐷𝑣) < 𝑥))
22	21	raleqbi1dv 3146	. . . . . 6 ⊢ (𝑠 = (𝐹 “ 𝑦) → (∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 (𝑢𝐷𝑣) < 𝑥 ↔ ∀𝑢 ∈ (𝐹 “ 𝑦)∀𝑣 ∈ (𝐹 “ 𝑦)(𝑢𝐷𝑣) < 𝑥))
23	20, 22	rexrnmpt 6369	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ∈ V → (∃𝑠 ∈ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 (𝑢𝐷𝑣) < 𝑥 ↔ ∃𝑦 ∈ 𝐵 ∀𝑢 ∈ (𝐹 “ 𝑦)∀𝑣 ∈ (𝐹 “ 𝑦)(𝑢𝐷𝑣) < 𝑥))
24	19, 23	syl 17	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑠 ∈ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 (𝑢𝐷𝑣) < 𝑥 ↔ ∃𝑦 ∈ 𝐵 ∀𝑢 ∈ (𝐹 “ 𝑦)∀𝑣 ∈ (𝐹 “ 𝑦)(𝑢𝐷𝑣) < 𝑥))
25		simpl3 1066	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 ∈ 𝐵) → 𝐹:𝑌⟶𝑋)
26		ffn 6045	. . . . . . . 8 ⊢ (𝐹:𝑌⟶𝑋 → 𝐹 Fn 𝑌)
27	25, 26	syl 17	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 ∈ 𝐵) → 𝐹 Fn 𝑌)
28		fbelss 21637	. . . . . . . 8 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ 𝑌)
29	6, 28	sylan 488	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ 𝑌)
30		oveq1 6657	. . . . . . . . . 10 ⊢ (𝑢 = (𝐹‘𝑧) → (𝑢𝐷𝑣) = ((𝐹‘𝑧)𝐷𝑣))
31	30	breq1d 4663	. . . . . . . . 9 ⊢ (𝑢 = (𝐹‘𝑧) → ((𝑢𝐷𝑣) < 𝑥 ↔ ((𝐹‘𝑧)𝐷𝑣) < 𝑥))
32	31	ralbidv 2986	. . . . . . . 8 ⊢ (𝑢 = (𝐹‘𝑧) → (∀𝑣 ∈ (𝐹 “ 𝑦)(𝑢𝐷𝑣) < 𝑥 ↔ ∀𝑣 ∈ (𝐹 “ 𝑦)((𝐹‘𝑧)𝐷𝑣) < 𝑥))
33	32	ralima 6498	. . . . . . 7 ⊢ ((𝐹 Fn 𝑌 ∧ 𝑦 ⊆ 𝑌) → (∀𝑢 ∈ (𝐹 “ 𝑦)∀𝑣 ∈ (𝐹 “ 𝑦)(𝑢𝐷𝑣) < 𝑥 ↔ ∀𝑧 ∈ 𝑦 ∀𝑣 ∈ (𝐹 “ 𝑦)((𝐹‘𝑧)𝐷𝑣) < 𝑥))
34	27, 29, 33	syl2anc 693	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 ∈ 𝐵) → (∀𝑢 ∈ (𝐹 “ 𝑦)∀𝑣 ∈ (𝐹 “ 𝑦)(𝑢𝐷𝑣) < 𝑥 ↔ ∀𝑧 ∈ 𝑦 ∀𝑣 ∈ (𝐹 “ 𝑦)((𝐹‘𝑧)𝐷𝑣) < 𝑥))
35		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑣 = (𝐹‘𝑤) → ((𝐹‘𝑧)𝐷𝑣) = ((𝐹‘𝑧)𝐷(𝐹‘𝑤)))
36	35	breq1d 4663	. . . . . . . . 9 ⊢ (𝑣 = (𝐹‘𝑤) → (((𝐹‘𝑧)𝐷𝑣) < 𝑥 ↔ ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥))
37	36	ralima 6498	. . . . . . . 8 ⊢ ((𝐹 Fn 𝑌 ∧ 𝑦 ⊆ 𝑌) → (∀𝑣 ∈ (𝐹 “ 𝑦)((𝐹‘𝑧)𝐷𝑣) < 𝑥 ↔ ∀𝑤 ∈ 𝑦 ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥))
38	27, 29, 37	syl2anc 693	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 ∈ 𝐵) → (∀𝑣 ∈ (𝐹 “ 𝑦)((𝐹‘𝑧)𝐷𝑣) < 𝑥 ↔ ∀𝑤 ∈ 𝑦 ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥))
39	38	ralbidv 2986	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 ∈ 𝐵) → (∀𝑧 ∈ 𝑦 ∀𝑣 ∈ (𝐹 “ 𝑦)((𝐹‘𝑧)𝐷𝑣) < 𝑥 ↔ ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥))
40	34, 39	bitrd 268	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 ∈ 𝐵) → (∀𝑢 ∈ (𝐹 “ 𝑦)∀𝑣 ∈ (𝐹 “ 𝑦)(𝑢𝐷𝑣) < 𝑥 ↔ ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥))
41	40	rexbidva 3049	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑦 ∈ 𝐵 ∀𝑢 ∈ (𝐹 “ 𝑦)∀𝑣 ∈ (𝐹 “ 𝑦)(𝑢𝐷𝑣) < 𝑥 ↔ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥))
42	24, 41	bitrd 268	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑠 ∈ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 (𝑢𝐷𝑣) < 𝑥 ↔ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥))
43	42	ralbidv 2986	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∀𝑥 ∈ ℝ+ ∃𝑠 ∈ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 (𝑢𝐷𝑣) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥))
44	4, 13, 43	3bitrd 294	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (((𝑋 FilMap 𝐹)‘𝐵) ∈ (CauFil‘𝐷) ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574   class class class wbr 4653   ↦ cmpt 4729  dom cdm 5114  ran crn 5115   “ cima 5117   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   < clt 10074  ℝ⁺crp 11832  ∞Metcxmt 19731  fBascfbas 19734  filGencfg 19735   FilMap cfm 21737  CauFilccfil 23050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-2 11079  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-xmet 19739  df-fbas 19743  df-fg 19744  df-fil 21650  df-fm 21742  df-cfil 23053

This theorem is referenced by:  caucfil  23081