Theorem metustrel 22357

Description: Elements of the filter base generated by the metric D are relations. (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
metustrel	|- ( ( D e. (PsMet ` X ) /\ A e. F ) -> Rel A )

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

Proof of Theorem metustrel

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		xpss 5226	. . 3 |- ( X X. X ) C_ ( _V X. _V )
4	2, 3	syl6ss 3615	. 2 |- ( ( D e. (PsMet ` X ) /\ A e. F ) -> A C_ ( _V X. _V ) )
5		df-rel 5121	. 2 |- ( Rel A <-> A C_ ( _V X. _V ) )
6	4, 5	sylibr 224	1 |- ( ( D e. (PsMet ` X ) /\ A e. F ) -> Rel A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   ran crn 5115   "cima 5117   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   0cc0 9936   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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-xr 10078  df-psmet 19738

This theorem is referenced by: (None)