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Theorem rnmpt 4610
Description: The range of a function in maps-to notation. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
rnmpt.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
rnmpt  |-  ran  F  =  { y  |  E. x  e.  A  y  =  B }
Distinct variable groups:    y, A    y, B    x, y
Allowed substitution hints:    A( x)    B( x)    F( x, y)

Proof of Theorem rnmpt
StepHypRef Expression
1 rnopab 4609 . 2  |-  ran  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }  =  { y  |  E. x ( x  e.  A  /\  y  =  B ) }
2 rnmpt.1 . . . 4  |-  F  =  ( x  e.  A  |->  B )
3 df-mpt 3848 . . . 4  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
42, 3eqtri 2076 . . 3  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
54rneqi 4590 . 2  |-  ran  F  =  ran  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
6 df-rex 2329 . . 3  |-  ( E. x  e.  A  y  =  B  <->  E. x
( x  e.  A  /\  y  =  B
) )
76abbii 2169 . 2  |-  { y  |  E. x  e.  A  y  =  B }  =  { y  |  E. x ( x  e.  A  /\  y  =  B ) }
81, 5, 73eqtr4i 2086 1  |-  ran  F  =  { y  |  E. x  e.  A  y  =  B }
Colors of variables: wff set class
Syntax hints:    /\ wa 101    = wceq 1259   E.wex 1397    e. wcel 1409   {cab 2042   E.wrex 2324   {copab 3845    |-> cmpt 3846   ran crn 4374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-mpt 3848  df-cnv 4381  df-dm 4383  df-rn 4384
This theorem is referenced by:  elrnmpt  4611  elrnmpt1  4613  elrnmptg  4614  dfiun3g  4617  dfiin3g  4618  fnrnfv  5248  fmpt  5347  fnasrn  5369  fnasrng  5371  fliftf  5467  abrexex  5772  abrexexg  5773  fo1st  5812  fo2nd  5813  qliftf  6222
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