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Theorem fnasrn 5718
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 5717 . 2  |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  { <. x ,  B >. }
3 eqid 2296 . . . . 5  |-  ( x  e.  A  |->  <. x ,  B >. )  =  ( x  e.  A  |->  <.
x ,  B >. )
43rnmpt 4941 . . . 4  |-  ran  (
x  e.  A  |->  <.
x ,  B >. )  =  { y  |  E. x  e.  A  y  =  <. x ,  B >. }
5 elsn 3668 . . . . . 6  |-  ( y  e.  { <. x ,  B >. }  <->  y  =  <. x ,  B >. )
65rexbii 2581 . . . . 5  |-  ( E. x  e.  A  y  e.  { <. x ,  B >. }  <->  E. x  e.  A  y  =  <. x ,  B >. )
76abbii 2408 . . . 4  |-  { y  |  E. x  e.  A  y  e.  { <. x ,  B >. } }  =  { y  |  E. x  e.  A  y  =  <. x ,  B >. }
84, 7eqtr4i 2319 . . 3  |-  ran  (
x  e.  A  |->  <.
x ,  B >. )  =  { y  |  E. x  e.  A  y  e.  { <. x ,  B >. } }
9 df-iun 3923 . . 3  |-  U_ x  e.  A  { <. x ,  B >. }  =  {
y  |  E. x  e.  A  y  e.  {
<. x ,  B >. } }
108, 9eqtr4i 2319 . 2  |-  ran  (
x  e.  A  |->  <.
x ,  B >. )  =  U_ x  e.  A  { <. x ,  B >. }
112, 10eqtr4i 2319 1  |-  ( x  e.  A  |->  B )  =  ran  ( x  e.  A  |->  <. x ,  B >. )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801   {csn 3653   <.cop 3656   U_ciun 3921    e. cmpt 4093   ran crn 4706
This theorem is referenced by:  resfunexg  5753  idref  5775  gruf  8449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278
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