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Theorem rnmpo 5978
Description: The range of an operation given by the maps-to notation. (Contributed by FL, 20-Jun-2011.)
Hypothesis
Ref Expression
rngop.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
rnmpo  |-  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
Distinct variable groups:    y, z, A   
z, B    z, C    z, F    x, y, z
Allowed substitution hints:    A( x)    B( x, y)    C( x, y)    F( x, y)

Proof of Theorem rnmpo
StepHypRef Expression
1 rngop.1 . . . 4  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
2 df-mpo 5873 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
31, 2eqtri 2198 . . 3  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
43rneqi 4850 . 2  |-  ran  F  =  ran  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
5 rnoprab2 5952 . 2  |-  ran  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
64, 5eqtri 2198 1  |-  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1353    e. wcel 2148   {cab 2163   E.wrex 2456   ran crn 4623   {coprab 5869    e. cmpo 5870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-cnv 4630  df-dm 4632  df-rn 4633  df-oprab 5872  df-mpo 5873
This theorem is referenced by:  elrnmpog  5980  elrnmpo  5981  ralrnmpo  5982  rexrnmpo  5983  mpoexw  6207  txuni2  13389  txbas  13391  txrest  13409
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