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Theorem rnmpt 4582
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 4581 . 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 3820 . . . 4  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
42, 3eqtri 2060 . . 3  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
54rneqi 4562 . 2  |-  ran  F  =  ran  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
6 df-rex 2312 . . 3  |-  ( E. x  e.  A  y  =  B  <->  E. x
( x  e.  A  /\  y  =  B
) )
76abbii 2153 . 2  |-  { y  |  E. x  e.  A  y  =  B }  =  { y  |  E. x ( x  e.  A  /\  y  =  B ) }
81, 5, 73eqtr4i 2070 1  |-  ran  F  =  { y  |  E. x  e.  A  y  =  B }
Colors of variables: wff set class
Syntax hints:    /\ wa 97    = wceq 1243   E.wex 1381    e. wcel 1393   {cab 2026   E.wrex 2307   {copab 3817    |-> cmpt 3818   ran crn 4346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-mpt 3820  df-cnv 4353  df-dm 4355  df-rn 4356
This theorem is referenced by:  elrnmpt  4583  elrnmpt1  4585  elrnmptg  4586  dfiun3g  4589  dfiin3g  4590  fnrnfv  5220  fmpt  5319  fnasrn  5341  fnasrng  5343  fliftf  5439  abrexex  5744  abrexexg  5745  fo1st  5784  fo2nd  5785  qliftf  6191
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