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Theorem elrnmpt2 6397
Description: Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
rngop.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
elrnmpt2.1  |-  C  e. 
_V
Assertion
Ref Expression
elrnmpt2  |-  ( D  e.  ran  F  <->  E. x  e.  A  E. y  e.  B  D  =  C )
Distinct variable groups:    y, A    x, y, D
Allowed substitution hints:    A( x)    B( x, y)    C( x, y)    F( x, y)

Proof of Theorem elrnmpt2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 rngop.1 . . . 4  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
21rnmpt2 6394 . . 3  |-  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
32eleq2i 2545 . 2  |-  ( D  e.  ran  F  <->  D  e.  { z  |  E. x  e.  A  E. y  e.  B  z  =  C } )
4 elrnmpt2.1 . . . . . 6  |-  C  e. 
_V
5 eleq1 2539 . . . . . 6  |-  ( D  =  C  ->  ( D  e.  _V  <->  C  e.  _V ) )
64, 5mpbiri 233 . . . . 5  |-  ( D  =  C  ->  D  e.  _V )
76rexlimivw 2952 . . . 4  |-  ( E. y  e.  B  D  =  C  ->  D  e. 
_V )
87rexlimivw 2952 . . 3  |-  ( E. x  e.  A  E. y  e.  B  D  =  C  ->  D  e. 
_V )
9 eqeq1 2471 . . . 4  |-  ( z  =  D  ->  (
z  =  C  <->  D  =  C ) )
1092rexbidv 2980 . . 3  |-  ( z  =  D  ->  ( E. x  e.  A  E. y  e.  B  z  =  C  <->  E. x  e.  A  E. y  e.  B  D  =  C ) )
118, 10elab3 3257 . 2  |-  ( D  e.  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }  <->  E. x  e.  A  E. y  e.  B  D  =  C )
123, 11bitri 249 1  |-  ( D  e.  ran  F  <->  E. x  e.  A  E. y  e.  B  D  =  C )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2815   _Vcvv 3113   ran crn 5000    |-> cmpt2 6284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-cnv 5007  df-dm 5009  df-rn 5010  df-oprab 6286  df-mpt2 6287
This theorem is referenced by:  qexALT  11193  lsmelvalx  16456  efgtlen  16540  frgpnabllem1  16668  fmucndlem  20529  mbfimaopnlem  21797  tglnunirn  23663  tpr2rico  27530  mbfmco2  27876  br2base  27880  dya2icobrsiga  27887  dya2iocnrect  27892  dya2iocucvr  27895  sxbrsigalem2  27897  cntotbnd  29895  eldiophb  30294
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