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Theorem dfmpt3 5072
Description: Alternate definition for the "maps to" notation df-mpt 3861. (Contributed by Mario Carneiro, 30-Dec-2016.)
Assertion
Ref Expression
dfmpt3  |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  ( { x }  X.  { B }
)

Proof of Theorem dfmpt3
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mpt 3861 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
2 velsn 3433 . . . . . . 7  |-  ( y  e.  { B }  <->  y  =  B )
32anbi2i 445 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  { B } )  <->  ( x  e.  A  /\  y  =  B ) )
43anbi2i 445 . . . . 5  |-  ( ( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  { B } ) )  <->  ( z  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  =  B ) ) )
542exbii 1538 . . . 4  |-  ( E. x E. y ( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  { B } ) )  <->  E. x E. y ( z  = 
<. x ,  y >.  /\  ( x  e.  A  /\  y  =  B
) ) )
6 eliunxp 4523 . . . 4  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  { B }
)  <->  E. x E. y
( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  { B } ) ) )
7 elopab 4041 . . . 4  |-  ( z  e.  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) } 
<->  E. x E. y
( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  =  B )
) )
85, 6, 73bitr4i 210 . . 3  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  { B }
)  <->  z  e.  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) } )
98eqriv 2080 . 2  |-  U_ x  e.  A  ( {
x }  X.  { B } )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
101, 9eqtr4i 2106 1  |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  ( { x }  X.  { B }
)
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285   E.wex 1422    e. wcel 1434   {csn 3416   <.cop 3419   U_ciun 3698   {copab 3858    |-> 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-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  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-opab 3860  df-mpt 3861  df-xp 4397  df-rel 4398
This theorem is referenced by:  dfmpt  5393  dfmptg  5395
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