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Theorem elrnmpt2 5973
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 5970 . . 3  |-  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
32eleq2i 2360 . 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 2356 . . . . . 6  |-  ( D  =  C  ->  ( D  e.  _V  <->  C  e.  _V ) )
64, 5mpbiri 224 . . . . 5  |-  ( D  =  C  ->  D  e.  _V )
76rexlimivw 2676 . . . 4  |-  ( E. y  e.  B  D  =  C  ->  D  e. 
_V )
87rexlimivw 2676 . . 3  |-  ( E. x  e.  A  E. y  e.  B  D  =  C  ->  D  e. 
_V )
9 eqeq1 2302 . . . 4  |-  ( z  =  D  ->  (
z  =  C  <->  D  =  C ) )
1092rexbidv 2599 . . 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 2934 . 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 240 1  |-  ( D  e.  ran  F  <->  E. x  e.  A  E. y  e.  B  D  =  C )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801   ran crn 4706    e. cmpt2 5876
This theorem is referenced by:  qexALT  10347  lsmelvalx  14967  efgtlen  15051  frgpnabllem1  15177  mbfimaopnlem  19026  tpr2rico  23311  mbfmco2  23585  br2base  23589  dya2iocrrnval  23597  iscst3  25279  cntotbnd  26623  eldiophb  26939
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-cnv 4713  df-dm 4715  df-rn 4716  df-oprab 5878  df-mpt2 5879
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