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Theorem dfmpt3 5294
Description: Alternate definition for the maps-to notation df-mpt 4029. (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 4029 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
2 velsn 3578 . . . . . . 7  |-  ( y  e.  { B }  <->  y  =  B )
32anbi2i 453 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  { B } )  <->  ( x  e.  A  /\  y  =  B ) )
43anbi2i 453 . . . . 5  |-  ( ( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  { B } ) )  <->  ( z  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  =  B ) ) )
542exbii 1586 . . . 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 4727 . . . 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 4220 . . . 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 2154 . 2  |-  U_ x  e.  A  ( {
x }  X.  { B } )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
101, 9eqtr4i 2181 1  |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  ( { x }  X.  { B }
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1335   E.wex 1472    e. wcel 2128   {csn 3561   <.cop 3564   U_ciun 3851   {copab 4026    |-> cmpt 4027    X. cxp 4586
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-iun 3853  df-opab 4028  df-mpt 4029  df-xp 4594  df-rel 4595
This theorem is referenced by:  dfmpt  5646  dfmptg  5648
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