Theorem elo1 14257

Description: Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)

Assertion

Ref	Expression
elo1	|- ( F e. O(1) <-> ( F e. ( CC ^pm RR ) /\ E. x e. RR E. m e. RR A. y e. ( dom F i^i ( x [,) +oo ) ) ( abs ` ( F ` y ) ) <_ m ) )

Distinct variable group:   x, m, y, F

Proof of Theorem elo1

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmeq 5324	. . . . 5 |- ( f = F -> dom f = dom F )
2	1	ineq1d 3813	. . . 4 |- ( f = F -> ( dom f i^i ( x [,) +oo ) ) = ( dom F i^i ( x [,) +oo ) ) )
3		fveq1 6190	. . . . . 6 |- ( f = F -> ( f ` y ) = ( F ` y ) )
4	3	fveq2d 6195	. . . . 5 |- ( f = F -> ( abs ` ( f ` y ) ) = ( abs ` ( F ` y ) ) )
5	4	breq1d 4663	. . . 4 |- ( f = F -> ( ( abs ` ( f ` y ) ) <_ m <-> ( abs ` ( F ` y ) ) <_ m ) )
6	2, 5	raleqbidv 3152	. . 3 |- ( f = F -> ( A. y e. ( dom f i^i ( x [,) +oo ) ) ( abs ` ( f ` y ) ) <_ m <-> A. y e. ( dom F i^i ( x [,) +oo ) ) ( abs ` ( F ` y ) ) <_ m ) )
7	6	2rexbidv 3057	. 2 |- ( f = F -> ( E. x e. RR E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( abs ` ( f ` y ) ) <_ m <-> E. x e. RR E. m e. RR A. y e. ( dom F i^i ( x [,) +oo ) ) ( abs ` ( F ` y ) ) <_ m ) )
8		df-o1 14221	. 2 |- O(1) = { f e. ( CC ^pm RR ) | E. x e. RR E. m e. RR A. y e. ( dom f i^i ( x [,) +oo ) ) ( abs ` ( f ` y ) ) <_ m }
9	7, 8	elrab2 3366	1 |- ( F e. O(1) <-> ( F e. ( CC ^pm RR ) /\ E. x e. RR E. m e. RR A. y e. ( dom F i^i ( x [,) +oo ) ) ( abs ` ( F ` y ) ) <_ m ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   RRcr 9935   +oocpnf 10071   <_ cle 10075   [,)cico 12177   abscabs 13974   O(1)co1 14217

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-dm 5124  df-iota 5851  df-fv 5896  df-o1 14221

This theorem is referenced by:  elo12  14258  o1f  14260  o1dm  14261