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Theorem mpt2mptx 5626
Description: Express a two-argument function as a one-argument function, or vice-versa. In this version 
B ( x ) is not assumed to be constant w.r.t  x. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypothesis
Ref Expression
mpt2mpt.1  |-  ( z  =  <. x ,  y
>.  ->  C  =  D )
Assertion
Ref Expression
mpt2mptx  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
Distinct variable groups:    x, y, z, A    y, B, z   
x, C, y    z, D
Allowed substitution hints:    B( x)    C( z)    D( x, y)

Proof of Theorem mpt2mptx
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 df-mpt 3849 . 2  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  C )  =  { <. z ,  w >.  |  ( z  e.  U_ x  e.  A  ( {
x }  X.  B
)  /\  w  =  C ) }
2 df-mpt2 5548 . . 3  |-  ( x  e.  A ,  y  e.  B  |->  D )  =  { <. <. x ,  y >. ,  w >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) }
3 eliunxp 4503 . . . . . . 7  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  <->  E. x E. y ( z  = 
<. x ,  y >.  /\  ( x  e.  A  /\  y  e.  B
) ) )
43anbi1i 446 . . . . . 6  |-  ( ( z  e.  U_ x  e.  A  ( {
x }  X.  B
)  /\  w  =  C )  <->  ( E. x E. y ( z  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  e.  B ) )  /\  w  =  C )
)
5 19.41vv 1825 . . . . . 6  |-  ( E. x E. y ( ( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  /\  w  =  C )  <->  ( E. x E. y ( z  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  e.  B ) )  /\  w  =  C )
)
6 anass 393 . . . . . . . 8  |-  ( ( ( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  /\  w  =  C )  <->  ( z  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  C ) ) )
7 mpt2mpt.1 . . . . . . . . . . 11  |-  ( z  =  <. x ,  y
>.  ->  C  =  D )
87eqeq2d 2093 . . . . . . . . . 10  |-  ( z  =  <. x ,  y
>.  ->  ( w  =  C  <->  w  =  D
) )
98anbi2d 452 . . . . . . . . 9  |-  ( z  =  <. x ,  y
>.  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  w  =  C
)  <->  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) ) )
109pm5.32i 442 . . . . . . . 8  |-  ( ( z  =  <. x ,  y >.  /\  (
( x  e.  A  /\  y  e.  B
)  /\  w  =  C ) )  <->  ( z  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) ) )
116, 10bitri 182 . . . . . . 7  |-  ( ( ( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  /\  w  =  C )  <->  ( z  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) ) )
12112exbii 1538 . . . . . 6  |-  ( E. x E. y ( ( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  /\  w  =  C )  <->  E. x E. y ( z  = 
<. x ,  y >.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) ) )
134, 5, 123bitr2i 206 . . . . 5  |-  ( ( z  e.  U_ x  e.  A  ( {
x }  X.  B
)  /\  w  =  C )  <->  E. x E. y ( z  = 
<. x ,  y >.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) ) )
1413opabbii 3853 . . . 4  |-  { <. z ,  w >.  |  ( z  e.  U_ x  e.  A  ( {
x }  X.  B
)  /\  w  =  C ) }  =  { <. z ,  w >.  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) ) }
15 dfoprab2 5583 . . . 4  |-  { <. <.
x ,  y >. ,  w >.  |  (
( x  e.  A  /\  y  e.  B
)  /\  w  =  D ) }  =  { <. z ,  w >.  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) ) }
1614, 15eqtr4i 2105 . . 3  |-  { <. z ,  w >.  |  ( z  e.  U_ x  e.  A  ( {
x }  X.  B
)  /\  w  =  C ) }  =  { <. <. x ,  y
>. ,  w >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) }
172, 16eqtr4i 2105 . 2  |-  ( x  e.  A ,  y  e.  B  |->  D )  =  { <. z ,  w >.  |  (
z  e.  U_ x  e.  A  ( {
x }  X.  B
)  /\  w  =  C ) }
181, 17eqtr4i 2105 1  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285   E.wex 1422    e. wcel 1434   {csn 3406   <.cop 3409   U_ciun 3686   {copab 3846    |-> cmpt 3847    X. cxp 4369   {coprab 5544    |-> cmpt2 5545
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 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-iun 3688  df-opab 3848  df-mpt 3849  df-xp 4377  df-rel 4378  df-oprab 5547  df-mpt2 5548
This theorem is referenced by:  mpt2mpt  5627  mpt2mptsx  5854  dmmpt2ssx  5856  fmpt2x  5857
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