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Theorem fnasrn 5702
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 5701 . 2  |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  { <. x ,  B >. }
3 eqid 2283 . . . . 5  |-  ( x  e.  A  |->  <. x ,  B >. )  =  ( x  e.  A  |->  <.
x ,  B >. )
43rnmpt 4925 . . . 4  |-  ran  (
x  e.  A  |->  <.
x ,  B >. )  =  { y  |  E. x  e.  A  y  =  <. x ,  B >. }
5 elsn 3655 . . . . . 6  |-  ( y  e.  { <. x ,  B >. }  <->  y  =  <. x ,  B >. )
65rexbii 2568 . . . . 5  |-  ( E. x  e.  A  y  e.  { <. x ,  B >. }  <->  E. x  e.  A  y  =  <. x ,  B >. )
76abbii 2395 . . . 4  |-  { y  |  E. x  e.  A  y  e.  { <. x ,  B >. } }  =  { y  |  E. x  e.  A  y  =  <. x ,  B >. }
84, 7eqtr4i 2306 . . 3  |-  ran  (
x  e.  A  |->  <.
x ,  B >. )  =  { y  |  E. x  e.  A  y  e.  { <. x ,  B >. } }
9 df-iun 3907 . . 3  |-  U_ x  e.  A  { <. x ,  B >. }  =  {
y  |  E. x  e.  A  y  e.  {
<. x ,  B >. } }
108, 9eqtr4i 2306 . 2  |-  ran  (
x  e.  A  |->  <.
x ,  B >. )  =  U_ x  e.  A  { <. x ,  B >. }
112, 10eqtr4i 2306 1  |-  ( x  e.  A  |->  B )  =  ran  ( x  e.  A  |->  <. x ,  B >. )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788   {csn 3640   <.cop 3643   U_ciun 3905    e. cmpt 4077   ran crn 4690
This theorem is referenced by:  resfunexg  5737  idref  5759  gruf  8433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262
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