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Theorem fnasrn 5566
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 5565 . 2  |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  { <. x ,  B >. }
3 eqid 2117 . . . . 5  |-  ( x  e.  A  |->  <. x ,  B >. )  =  ( x  e.  A  |->  <.
x ,  B >. )
43rnmpt 4757 . . . 4  |-  ran  (
x  e.  A  |->  <.
x ,  B >. )  =  { y  |  E. x  e.  A  y  =  <. x ,  B >. }
5 velsn 3514 . . . . . 6  |-  ( y  e.  { <. x ,  B >. }  <->  y  =  <. x ,  B >. )
65rexbii 2419 . . . . 5  |-  ( E. x  e.  A  y  e.  { <. x ,  B >. }  <->  E. x  e.  A  y  =  <. x ,  B >. )
76abbii 2233 . . . 4  |-  { y  |  E. x  e.  A  y  e.  { <. x ,  B >. } }  =  { y  |  E. x  e.  A  y  =  <. x ,  B >. }
84, 7eqtr4i 2141 . . 3  |-  ran  (
x  e.  A  |->  <.
x ,  B >. )  =  { y  |  E. x  e.  A  y  e.  { <. x ,  B >. } }
9 df-iun 3785 . . 3  |-  U_ x  e.  A  { <. x ,  B >. }  =  {
y  |  E. x  e.  A  y  e.  {
<. x ,  B >. } }
108, 9eqtr4i 2141 . 2  |-  ran  (
x  e.  A  |->  <.
x ,  B >. )  =  U_ x  e.  A  { <. x ,  B >. }
112, 10eqtr4i 2141 1  |-  ( x  e.  A  |->  B )  =  ran  ( x  e.  A  |->  <. x ,  B >. )
Colors of variables: wff set class
Syntax hints:    = wceq 1316    e. wcel 1465   {cab 2103   E.wrex 2394   _Vcvv 2660   {csn 3497   <.cop 3500   U_ciun 3783    |-> cmpt 3959   ran crn 4510
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100
This theorem is referenced by:  idref  5626
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