Theorem metidval 29933

Description: Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)

Assertion

Ref	Expression
metidval	|- ( D e. (PsMet ` X ) -> (~Met ` D ) = { <. x , y >. | ( ( x e. X /\ y e. X ) /\ ( x D y ) = 0 ) } )

Distinct variable groups:   x, y, D   x, X, y

Proof of Theorem metidval

Dummy variables w d z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-metid 29931	. . 3 |- ~Met = ( d e. U. ran PsMet |-> { <. x , y >. | ( ( x e. dom dom d /\ y e. dom dom d ) /\ ( x d y ) = 0 ) } )
2	1	a1i 11	. 2 |- ( D e. (PsMet ` X ) -> ~Met = ( d e. U. ran PsMet |-> { <. x , y >. | ( ( x e. dom dom d /\ y e. dom dom d ) /\ ( x d y ) = 0 ) } ) )
3		simpr 477	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> d = D )
4	3	dmeqd 5326	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> dom d = dom D )
5	4	dmeqd 5326	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> dom dom d = dom dom D )
6		psmetdmdm 22110	. . . . . . . 8 |- ( D e. (PsMet ` X ) -> X = dom dom D )
7	6	adantr 481	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> X = dom dom D )
8	5, 7	eqtr4d 2659	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> dom dom d = X )
9	8	eleq2d 2687	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( x e. dom dom d <-> x e. X ) )
10	8	eleq2d 2687	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( y e. dom dom d <-> y e. X ) )
11	9, 10	anbi12d 747	. . . 4 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( ( x e. dom dom d /\ y e. dom dom d ) <-> ( x e. X /\ y e. X ) ) )
12	3	oveqd 6667	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( x d y ) = ( x D y ) )
13	12	eqeq1d 2624	. . . 4 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( ( x d y ) = 0 <-> ( x D y ) = 0 ) )
14	11, 13	anbi12d 747	. . 3 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( ( ( x e. dom dom d /\ y e. dom dom d ) /\ ( x d y ) = 0 ) <-> ( ( x e. X /\ y e. X ) /\ ( x D y ) = 0 ) ) )
15	14	opabbidv 4716	. 2 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> { <. x , y >. | ( ( x e. dom dom d /\ y e. dom dom d ) /\ ( x d y ) = 0 ) } = { <. x , y >. | ( ( x e. X /\ y e. X ) /\ ( x D y ) = 0 ) } )
16		elfvdm 6220	. . . 4 |- ( D e. (PsMet ` X ) -> X e. dom PsMet )
17		fveq2 6191	. . . . . 6 |- ( x = X -> (PsMet ` x ) = (PsMet ` X ) )
18	17	eleq2d 2687	. . . . 5 |- ( x = X -> ( D e. (PsMet ` x ) <-> D e. (PsMet ` X ) ) )
19	18	rspcev 3309	. . . 4 |- ( ( X e. dom PsMet /\ D e. (PsMet ` X ) ) -> E. x e. dom PsMet D e. (PsMet ` x ) )
20	16, 19	mpancom 703	. . 3 |- ( D e. (PsMet ` X ) -> E. x e. dom PsMet D e. (PsMet ` x ) )
21		df-psmet 19738	. . . . 5 |- PsMet = ( x e. _V |-> { d e. ( RR* ^m ( x X. x ) ) | A. y e. x ( ( y d y ) = 0 /\ A. z e. x A. w e. x ( y d z ) <_ ( ( w d y ) +e ( w d z ) ) ) } )
22	21	funmpt2 5927	. . . 4 |- Fun PsMet
23		elunirn 6509	. . . 4 |- ( Fun PsMet -> ( D e. U. ran PsMet <-> E. x e. dom PsMet D e. (PsMet ` x ) ) )
24	22, 23	ax-mp 5	. . 3 |- ( D e. U. ran PsMet <-> E. x e. dom PsMet D e. (PsMet ` x ) )
25	20, 24	sylibr 224	. 2 |- ( D e. (PsMet ` X ) -> D e. U. ran PsMet )
26		opabssxp 5193	. . 3 |- { <. x , y >. | ( ( x e. X /\ y e. X ) /\ ( x D y ) = 0 ) } C_ ( X X. X )
27		elfvex 6221	. . . 4 |- ( D e. (PsMet ` X ) -> X e. _V )
28		xpexg 6960	. . . 4 |- ( ( X e. _V /\ X e. _V ) -> ( X X. X ) e. _V )
29	27, 27, 28	syl2anc 693	. . 3 |- ( D e. (PsMet ` X ) -> ( X X. X ) e. _V )
30		ssexg 4804	. . 3 |- ( ( { <. x , y >. | ( ( x e. X /\ y e. X ) /\ ( x D y ) = 0 ) } C_ ( X X. X ) /\ ( X X. X ) e. _V ) -> { <. x , y >. | ( ( x e. X /\ y e. X ) /\ ( x D y ) = 0 ) } e. _V )
31	26, 29, 30	sylancr 695	. 2 |- ( D e. (PsMet ` X ) -> { <. x , y >. | ( ( x e. X /\ y e. X ) /\ ( x D y ) = 0 ) } e. _V )
32	2, 15, 25, 31	fvmptd 6288	1 |- ( D e. (PsMet ` X ) -> (~Met ` D ) = { <. x , y >. | ( ( x e. X /\ y e. X ) /\ ( x D y ) = 0 ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   U.cuni 4436   class class class wbr 4653   {copab 4712   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   0cc0 9936   RR*cxr 10073   <_ cle 10075   +ecxad 11944  PsMetcpsmet 19730  ~Metcmetid 29929

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-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-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  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  df-metid 29931

This theorem is referenced by:  metidss  29934  metidv  29935