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Theorem dfmpt 5590
Description: Alternate definition for the maps-to notation df-mpt 3986 (although it requires that  B be a set). (Contributed by NM, 24-Aug-2010.) (Revised by Mario Carneiro, 30-Dec-2016.)
Hypothesis
Ref Expression
dfmpt.1  |-  B  e. 
_V
Assertion
Ref Expression
dfmpt  |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  { <. x ,  B >. }

Proof of Theorem dfmpt
StepHypRef Expression
1 dfmpt3 5240 . 2  |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  ( { x }  X.  { B }
)
2 vex 2684 . . . . 5  |-  x  e. 
_V
3 dfmpt.1 . . . . 5  |-  B  e. 
_V
42, 3xpsn 5589 . . . 4  |-  ( { x }  X.  { B } )  =  { <. x ,  B >. }
54a1i 9 . . 3  |-  ( x  e.  A  ->  ( { x }  X.  { B } )  =  { <. x ,  B >. } )
65iuneq2i 3826 . 2  |-  U_ x  e.  A  ( {
x }  X.  { B } )  =  U_ x  e.  A  { <. x ,  B >. }
71, 6eqtri 2158 1  |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  { <. x ,  B >. }
Colors of variables: wff set class
Syntax hints:    = wceq 1331    e. wcel 1480   _Vcvv 2681   {csn 3522   <.cop 3525   U_ciun 3808    |-> cmpt 3984    X. cxp 4532
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125
This theorem is referenced by:  fnasrn  5591  dfmpo  6113
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