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Theorem dfmpt3 5382
Description: Alternate definition for the "maps to" notation df-mpt 4095. (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 4095 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
2 elsn 3668 . . . . . . 7  |-  ( y  e.  { B }  <->  y  =  B )
32anbi2i 675 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  { B } )  <->  ( x  e.  A  /\  y  =  B ) )
43anbi2i 675 . . . . 5  |-  ( ( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  { B } ) )  <->  ( z  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  =  B ) ) )
542exbii 1573 . . . 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 4839 . . . 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 4288 . . . 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 268 . . 3  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  { B }
)  <->  z  e.  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) } )
98eqriv 2293 . 2  |-  U_ x  e.  A  ( {
x }  X.  { B } )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
101, 9eqtr4i 2319 1  |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  ( { x }  X.  { B }
)
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   {csn 3653   <.cop 3656   U_ciun 3921   {copab 4092    e. cmpt 4093    X. cxp 4703
This theorem is referenced by:  dfmpt  5717  taylpfval  19760  indval2  23613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-iun 3923  df-opab 4094  df-mpt 4095  df-xp 4711  df-rel 4712
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