Theorem metustsym 22360

Description: Elements of the filter base generated by the metric D are symmetric. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)

Hypothesis

Ref	Expression
metust.1	|- F = ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )

Assertion

Ref	Expression
metustsym	|- ( ( D e. (PsMet ` X ) /\ A e. F ) -> `' A = A )

Distinct variable groups:   D, a   X, a   A, a   F, a

Proof of Theorem metustsym

Dummy variables p q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		metust.1	. . . 4 |- F = ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )
2	1	metustss 22356	. . 3 |- ( ( D e. (PsMet ` X ) /\ A e. F ) -> A C_ ( X X. X ) )
3		cnvss 5294	. . . 4 |- ( A C_ ( X X. X ) -> `' A C_ `' ( X X. X ) )
4		cnvxp 5551	. . . 4 |- `' ( X X. X ) = ( X X. X )
5	3, 4	syl6sseq 3651	. . 3 |- ( A C_ ( X X. X ) -> `' A C_ ( X X. X ) )
6	2, 5	syl 17	. 2 |- ( ( D e. (PsMet ` X ) /\ A e. F ) -> `' A C_ ( X X. X ) )
7		simp-4l 806	. . . . . . . . . 10 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> D e. (PsMet ` X ) )
8		simpr1r 1119	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( ( p e. X /\ q e. X ) /\ a e. RR+ /\ A = ( `' D " ( 0 [,) a ) ) ) ) -> q e. X )
9	8	3anassrs 1290	. . . . . . . . . 10 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> q e. X )
10		simpr1l 1118	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( ( p e. X /\ q e. X ) /\ a e. RR+ /\ A = ( `' D " ( 0 [,) a ) ) ) ) -> p e. X )
11	10	3anassrs 1290	. . . . . . . . . 10 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> p e. X )
12		psmetsym 22115	. . . . . . . . . 10 |- ( ( D e. (PsMet ` X ) /\ q e. X /\ p e. X ) -> ( q D p ) = ( p D q ) )
13	7, 9, 11, 12	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> ( q D p ) = ( p D q ) )
14		df-ov 6653	. . . . . . . . 9 |- ( q D p ) = ( D ` <. q , p >. )
15		df-ov 6653	. . . . . . . . 9 |- ( p D q ) = ( D ` <. p , q >. )
16	13, 14, 15	3eqtr3g 2679	. . . . . . . 8 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> ( D ` <. q , p >. ) = ( D ` <. p , q >. ) )
17	16	eleq1d 2686	. . . . . . 7 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> ( ( D ` <. q , p >. ) e. ( 0 [,) a ) <-> ( D ` <. p , q >. ) e. ( 0 [,) a ) ) )
18		psmetf 22111	. . . . . . . . 9 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
19		ffun 6048	. . . . . . . . 9 |- ( D : ( X X. X ) --> RR* -> Fun D )
20	7, 18, 19	3syl 18	. . . . . . . 8 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> Fun D )
21		simpllr 799	. . . . . . . . . . 11 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> ( p e. X /\ q e. X ) )
22	21	ancomd 467	. . . . . . . . . 10 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> ( q e. X /\ p e. X ) )
23		opelxpi 5148	. . . . . . . . . 10 |- ( ( q e. X /\ p e. X ) -> <. q , p >. e. ( X X. X ) )
24	22, 23	syl 17	. . . . . . . . 9 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> <. q , p >. e. ( X X. X ) )
25		fdm 6051	. . . . . . . . . 10 |- ( D : ( X X. X ) --> RR* -> dom D = ( X X. X ) )
26	7, 18, 25	3syl 18	. . . . . . . . 9 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> dom D = ( X X. X ) )
27	24, 26	eleqtrrd 2704	. . . . . . . 8 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> <. q , p >. e. dom D )
28		fvimacnv 6332	. . . . . . . 8 |- ( ( Fun D /\ <. q , p >. e. dom D ) -> ( ( D ` <. q , p >. ) e. ( 0 [,) a ) <-> <. q , p >. e. ( `' D " ( 0 [,) a ) ) ) )
29	20, 27, 28	syl2anc 693	. . . . . . 7 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> ( ( D ` <. q , p >. ) e. ( 0 [,) a ) <-> <. q , p >. e. ( `' D " ( 0 [,) a ) ) ) )
30		opelxpi 5148	. . . . . . . . . 10 |- ( ( p e. X /\ q e. X ) -> <. p , q >. e. ( X X. X ) )
31	21, 30	syl 17	. . . . . . . . 9 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> <. p , q >. e. ( X X. X ) )
32	31, 26	eleqtrrd 2704	. . . . . . . 8 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> <. p , q >. e. dom D )
33		fvimacnv 6332	. . . . . . . 8 |- ( ( Fun D /\ <. p , q >. e. dom D ) -> ( ( D ` <. p , q >. ) e. ( 0 [,) a ) <-> <. p , q >. e. ( `' D " ( 0 [,) a ) ) ) )
34	20, 32, 33	syl2anc 693	. . . . . . 7 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> ( ( D ` <. p , q >. ) e. ( 0 [,) a ) <-> <. p , q >. e. ( `' D " ( 0 [,) a ) ) ) )
35	17, 29, 34	3bitr3d 298	. . . . . 6 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> ( <. q , p >. e. ( `' D " ( 0 [,) a ) ) <-> <. p , q >. e. ( `' D " ( 0 [,) a ) ) ) )
36		simpr 477	. . . . . . 7 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> A = ( `' D " ( 0 [,) a ) ) )
37	36	eleq2d 2687	. . . . . 6 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> ( <. q , p >. e. A <-> <. q , p >. e. ( `' D " ( 0 [,) a ) ) ) )
38	36	eleq2d 2687	. . . . . 6 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> ( <. p , q >. e. A <-> <. p , q >. e. ( `' D " ( 0 [,) a ) ) ) )
39	35, 37, 38	3bitr4d 300	. . . . 5 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> ( <. q , p >. e. A <-> <. p , q >. e. A ) )
40		eqid 2622	. . . . . . . . 9 |- ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) = ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )
41	40	elrnmpt 5372	. . . . . . . 8 |- ( A e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) -> ( A e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) <-> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) ) )
42	41	ibi 256	. . . . . . 7 |- ( A e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) -> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) )
43	42, 1	eleq2s 2719	. . . . . 6 |- ( A e. F -> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) )
44	43	ad2antlr 763	. . . . 5 |- ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) -> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) )
45	39, 44	r19.29a 3078	. . . 4 |- ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) -> ( <. q , p >. e. A <-> <. p , q >. e. A ) )
46		df-br 4654	. . . . 5 |- ( p `' A q <-> <. p , q >. e. `' A )
47		vex 3203	. . . . . 6 |- p e. _V
48		vex 3203	. . . . . 6 |- q e. _V
49	47, 48	opelcnv 5304	. . . . 5 |- ( <. p , q >. e. `' A <-> <. q , p >. e. A )
50	46, 49	bitri 264	. . . 4 |- ( p `' A q <-> <. q , p >. e. A )
51		df-br 4654	. . . 4 |- ( p A q <-> <. p , q >. e. A )
52	45, 50, 51	3bitr4g 303	. . 3 |- ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ ( p e. X /\ q e. X ) ) -> ( p `' A q <-> p A q ) )
53	52	3impb 1260	. 2 |- ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ p e. X /\ q e. X ) -> ( p `' A q <-> p A q ) )
54	6, 2, 53	eqbrrdva 5291	1 |- ( ( D e. (PsMet ` X ) /\ A e. F ) -> `' A = A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   RR*cxr 10073   RR+crp 11832   [,)cico 12177  PsMetcpsmet 19730

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-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

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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655  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-xadd 11947  df-psmet 19738

This theorem is referenced by:  metust  22363