ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rnmpt2 Unicode version

Theorem rnmpt2 5663
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
rnmpt2  |-  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 rnmpt2
StepHypRef Expression
1 rngop.1 . . . 4  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
2 df-mpt2 5569 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
31, 2eqtri 2103 . . 3  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
43rneqi 4611 . 2  |-  ran  F  =  ran  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
5 rnoprab2 5640 . 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 2103 1  |-  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285    e. wcel 1434   {cab 2069   E.wrex 2354   ran crn 4393   {coprab 5565    |-> cmpt2 5566
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2612  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-cnv 4400  df-dm 4402  df-rn 4403  df-oprab 5568  df-mpt2 5569
This theorem is referenced by:  elrnmpt2g  5665  elrnmpt2  5666  ralrnmpt2  5667  rexrnmpt2  5668
  Copyright terms: Public domain W3C validator