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Theorem dfmpt3 5122
Description: Alternate definition for the maps-to notation df-mpt 3893. (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 3893 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
2 velsn 3458 . . . . . . 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 1542 . . . 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 4563 . . . 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 4076 . . . 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 2085 . 2  |-  U_ x  e.  A  ( {
x }  X.  { B } )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
101, 9eqtr4i 2111 1  |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  ( { x }  X.  { B }
)
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438   {csn 3441   <.cop 3444   U_ciun 3725   {copab 3890    |-> cmpt 3891    X. cxp 4426
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-iun 3727  df-opab 3892  df-mpt 3893  df-xp 4434  df-rel 4435
This theorem is referenced by:  dfmpt  5458  dfmptg  5460
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