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Theorem dfmpt 5851
Description: Alternate definition for the "maps to" notation df-mpt 4210 (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 5508 . 2  |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  ( { x }  X.  { B }
)
2 vex 2903 . . . . 5  |-  x  e. 
_V
3 dfmpt.1 . . . . 5  |-  B  e. 
_V
42, 3xpsn 5850 . . . 4  |-  ( { x }  X.  { B } )  =  { <. x ,  B >. }
54a1i 11 . . 3  |-  ( x  e.  A  ->  ( { x }  X.  { B } )  =  { <. x ,  B >. } )
65iuneq2i 4054 . 2  |-  U_ x  e.  A  ( {
x }  X.  { B } )  =  U_ x  e.  A  { <. x ,  B >. }
71, 6eqtri 2408 1  |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  { <. x ,  B >. }
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717   _Vcvv 2900   {csn 3758   <.cop 3761   U_ciun 4036    e. cmpt 4208    X. cxp 4817
This theorem is referenced by:  fnasrn  5852  dfmpt2  6377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402
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