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Theorem fnasrn 5914
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 5913 . 2  |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  { <. x ,  B >. }
3 eqid 2438 . . . . 5  |-  ( x  e.  A  |->  <. x ,  B >. )  =  ( x  e.  A  |->  <.
x ,  B >. )
43rnmpt 5118 . . . 4  |-  ran  (
x  e.  A  |->  <.
x ,  B >. )  =  { y  |  E. x  e.  A  y  =  <. x ,  B >. }
5 elsn 3831 . . . . . 6  |-  ( y  e.  { <. x ,  B >. }  <->  y  =  <. x ,  B >. )
65rexbii 2732 . . . . 5  |-  ( E. x  e.  A  y  e.  { <. x ,  B >. }  <->  E. x  e.  A  y  =  <. x ,  B >. )
76abbii 2550 . . . 4  |-  { y  |  E. x  e.  A  y  e.  { <. x ,  B >. } }  =  { y  |  E. x  e.  A  y  =  <. x ,  B >. }
84, 7eqtr4i 2461 . . 3  |-  ran  (
x  e.  A  |->  <.
x ,  B >. )  =  { y  |  E. x  e.  A  y  e.  { <. x ,  B >. } }
9 df-iun 4097 . . 3  |-  U_ x  e.  A  { <. x ,  B >. }  =  {
y  |  E. x  e.  A  y  e.  {
<. x ,  B >. } }
108, 9eqtr4i 2461 . 2  |-  ran  (
x  e.  A  |->  <.
x ,  B >. )  =  U_ x  e.  A  { <. x ,  B >. }
112, 10eqtr4i 2461 1  |-  ( x  e.  A  |->  B )  =  ran  ( x  e.  A  |->  <. x ,  B >. )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   {cab 2424   E.wrex 2708   _Vcvv 2958   {csn 3816   <.cop 3819   U_ciun 4095    e. cmpt 4268   ran crn 4881
This theorem is referenced by:  resfunexg  5959  idref  5981  gruf  8688
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463
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