ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfmpt Unicode version

Theorem dfmpt 5392
Description: Alternate definition for the "maps to" notation df-mpt 3861 (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 5072 . 2  |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  ( { x }  X.  { B }
)
2 vex 2613 . . . . 5  |-  x  e. 
_V
3 dfmpt.1 . . . . 5  |-  B  e. 
_V
42, 3xpsn 5391 . . . 4  |-  ( { x }  X.  { B } )  =  { <. x ,  B >. }
54a1i 9 . . 3  |-  ( x  e.  A  ->  ( { x }  X.  { B } )  =  { <. x ,  B >. } )
65iuneq2i 3716 . 2  |-  U_ x  e.  A  ( {
x }  X.  { B } )  =  U_ x  e.  A  { <. x ,  B >. }
71, 6eqtri 2103 1  |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  { <. x ,  B >. }
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434   _Vcvv 2610   {csn 3416   <.cop 3419   U_ciun 3698    |-> cmpt 3859    X. cxp 4389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-v 2612  df-sbc 2825  df-csb 2918  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959
This theorem is referenced by:  fnasrn  5393  dfmpt2  5895
  Copyright terms: Public domain W3C validator