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Theorem elrnmpt2 6176
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 6173 . . 3  |-  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
32eleq2i 2500 . 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 2496 . . . . . 6  |-  ( D  =  C  ->  ( D  e.  _V  <->  C  e.  _V ) )
64, 5mpbiri 225 . . . . 5  |-  ( D  =  C  ->  D  e.  _V )
76rexlimivw 2819 . . . 4  |-  ( E. y  e.  B  D  =  C  ->  D  e. 
_V )
87rexlimivw 2819 . . 3  |-  ( E. x  e.  A  E. y  e.  B  D  =  C  ->  D  e. 
_V )
9 eqeq1 2442 . . . 4  |-  ( z  =  D  ->  (
z  =  C  <->  D  =  C ) )
1092rexbidv 2741 . . 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 3082 . 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 241 1  |-  ( D  e.  ran  F  <->  E. x  e.  A  E. y  e.  B  D  =  C )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    e. wcel 1725   {cab 2422   E.wrex 2699   _Vcvv 2949   ran crn 4872    e. cmpt2 6076
This theorem is referenced by:  qexALT  10582  lsmelvalx  15267  efgtlen  15351  frgpnabllem1  15477  fmucndlem  18314  mbfimaopnlem  19540  tpr2rico  24303  mbfmco2  24608  br2base  24612  dya2icobrsiga  24619  dya2iocnrect  24624  dya2iocucvr  24627  sxbrsigalem2  24629  cntotbnd  26497  eldiophb  26807
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-br 4206  df-opab 4260  df-cnv 4879  df-dm 4881  df-rn 4882  df-oprab 6078  df-mpt2 6079
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