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Theorem dfmpt3 5213
Description: Alternate definition for the maps-to notation df-mpt 3959. (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 3959 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
2 velsn 3512 . . . . . . 7  |-  ( y  e.  { B }  <->  y  =  B )
32anbi2i 450 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  { B } )  <->  ( x  e.  A  /\  y  =  B ) )
43anbi2i 450 . . . . 5  |-  ( ( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  { B } ) )  <->  ( z  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  =  B ) ) )
542exbii 1568 . . . 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 4646 . . . 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 4148 . . . 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 211 . . 3  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  { B }
)  <->  z  e.  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) } )
98eqriv 2112 . 2  |-  U_ x  e.  A  ( {
x }  X.  { B } )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
101, 9eqtr4i 2139 1  |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  ( { x }  X.  { B }
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1314   E.wex 1451    e. wcel 1463   {csn 3495   <.cop 3498   U_ciun 3781   {copab 3956    |-> cmpt 3957    X. cxp 4505
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-iun 3783  df-opab 3958  df-mpt 3959  df-xp 4513  df-rel 4514
This theorem is referenced by:  dfmpt  5563  dfmptg  5565
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