Theorem isbnd 33579

Description: The predicate "is a bounded metric space". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)

Assertion

Ref	Expression
isbnd	|- ( M e. ( Bnd ` X ) <-> ( M e. ( Met ` X ) /\ A. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) ) )

Distinct variable groups:   x, r, M   X, r, x

Proof of Theorem isbnd

Dummy variables m y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6221	. 2 |- ( M e. ( Bnd ` X ) -> X e. _V )
2		elfvex 6221	. . 3 |- ( M e. ( Met ` X ) -> X e. _V )
3	2	adantr 481	. 2 |- ( ( M e. ( Met ` X ) /\ A. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) ) -> X e. _V )
4		fveq2 6191	. . . . . 6 |- ( y = X -> ( Met ` y ) = ( Met ` X ) )
5		eqeq1 2626	. . . . . . . 8 |- ( y = X -> ( y = ( x ( ball ` m ) r ) <-> X = ( x ( ball ` m ) r ) ) )
6	5	rexbidv 3052	. . . . . . 7 |- ( y = X -> ( E. r e. RR+ y = ( x ( ball ` m ) r ) <-> E. r e. RR+ X = ( x ( ball ` m ) r ) ) )
7	6	raleqbi1dv 3146	. . . . . 6 |- ( y = X -> ( A. x e. y E. r e. RR+ y = ( x ( ball ` m ) r ) <-> A. x e. X E. r e. RR+ X = ( x ( ball ` m ) r ) ) )
8	4, 7	rabeqbidv 3195	. . . . 5 |- ( y = X -> { m e. ( Met ` y ) | A. x e. y E. r e. RR+ y = ( x ( ball ` m ) r ) } = { m e. ( Met ` X ) | A. x e. X E. r e. RR+ X = ( x ( ball ` m ) r ) } )
9		df-bnd 33578	. . . . 5 |- Bnd = ( y e. _V |-> { m e. ( Met ` y ) | A. x e. y E. r e. RR+ y = ( x ( ball ` m ) r ) } )
10		fvex 6201	. . . . . 6 |- ( Met ` X ) e. _V
11	10	rabex 4813	. . . . 5 |- { m e. ( Met ` X ) | A. x e. X E. r e. RR+ X = ( x ( ball ` m ) r ) } e. _V
12	8, 9, 11	fvmpt 6282	. . . 4 |- ( X e. _V -> ( Bnd ` X ) = { m e. ( Met ` X ) | A. x e. X E. r e. RR+ X = ( x ( ball ` m ) r ) } )
13	12	eleq2d 2687	. . 3 |- ( X e. _V -> ( M e. ( Bnd ` X ) <-> M e. { m e. ( Met ` X ) | A. x e. X E. r e. RR+ X = ( x ( ball ` m ) r ) } ) )
14		fveq2 6191	. . . . . . . 8 |- ( m = M -> ( ball ` m ) = ( ball ` M ) )
15	14	oveqd 6667	. . . . . . 7 |- ( m = M -> ( x ( ball ` m ) r ) = ( x ( ball ` M ) r ) )
16	15	eqeq2d 2632	. . . . . 6 |- ( m = M -> ( X = ( x ( ball ` m ) r ) <-> X = ( x ( ball ` M ) r ) ) )
17	16	rexbidv 3052	. . . . 5 |- ( m = M -> ( E. r e. RR+ X = ( x ( ball ` m ) r ) <-> E. r e. RR+ X = ( x ( ball ` M ) r ) ) )
18	17	ralbidv 2986	. . . 4 |- ( m = M -> ( A. x e. X E. r e. RR+ X = ( x ( ball ` m ) r ) <-> A. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) ) )
19	18	elrab 3363	. . 3 |- ( M e. { m e. ( Met ` X ) | A. x e. X E. r e. RR+ X = ( x ( ball ` m ) r ) } <-> ( M e. ( Met ` X ) /\ A. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) ) )
20	13, 19	syl6bb 276	. 2 |- ( X e. _V -> ( M e. ( Bnd ` X ) <-> ( M e. ( Met ` X ) /\ A. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) ) ) )
21	1, 3, 20	pm5.21nii 368	1 |- ( M e. ( Bnd ` X ) <-> ( M e. ( Met ` X ) /\ A. x e. X E. r e. RR+ X = ( x ( ball ` M ) r ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   ` cfv 5888  (class class class)co 6650   RR+crp 11832   Metcme 19732   ballcbl 19733   Bndcbnd 33566

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

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-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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-bnd 33578

This theorem is referenced by:  bndmet  33580  isbndx  33581  isbnd3  33583  bndss  33585  totbndbnd  33588