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Theorem fnasrn 5564
Description: A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
dfmpt.1  |-  B  e. 
_V
Assertion
Ref Expression
fnasrn  |-  ( x  e.  A  |->  B )  =  ran  ( x  e.  A  |->  <. x ,  B >. )

Proof of Theorem fnasrn
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfmpt.1 . . 3  |-  B  e. 
_V
21dfmpt 5563 . 2  |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  { <. x ,  B >. }
3 eqid 2115 . . . . 5  |-  ( x  e.  A  |->  <. x ,  B >. )  =  ( x  e.  A  |->  <.
x ,  B >. )
43rnmpt 4755 . . . 4  |-  ran  (
x  e.  A  |->  <.
x ,  B >. )  =  { y  |  E. x  e.  A  y  =  <. x ,  B >. }
5 velsn 3512 . . . . . 6  |-  ( y  e.  { <. x ,  B >. }  <->  y  =  <. x ,  B >. )
65rexbii 2417 . . . . 5  |-  ( E. x  e.  A  y  e.  { <. x ,  B >. }  <->  E. x  e.  A  y  =  <. x ,  B >. )
76abbii 2231 . . . 4  |-  { y  |  E. x  e.  A  y  e.  { <. x ,  B >. } }  =  { y  |  E. x  e.  A  y  =  <. x ,  B >. }
84, 7eqtr4i 2139 . . 3  |-  ran  (
x  e.  A  |->  <.
x ,  B >. )  =  { y  |  E. x  e.  A  y  e.  { <. x ,  B >. } }
9 df-iun 3783 . . 3  |-  U_ x  e.  A  { <. x ,  B >. }  =  {
y  |  E. x  e.  A  y  e.  {
<. x ,  B >. } }
108, 9eqtr4i 2139 . 2  |-  ran  (
x  e.  A  |->  <.
x ,  B >. )  =  U_ x  e.  A  { <. x ,  B >. }
112, 10eqtr4i 2139 1  |-  ( x  e.  A  |->  B )  =  ran  ( x  e.  A  |->  <. x ,  B >. )
Colors of variables: wff set class
Syntax hints:    = wceq 1314    e. wcel 1463   {cab 2101   E.wrex 2392   _Vcvv 2658   {csn 3495   <.cop 3498   U_ciun 3781    |-> cmpt 3957   ran crn 4508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-reu 2398  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098
This theorem is referenced by:  idref  5624
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