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Theorem mpt2mptx2 30736
Description: Express a two-argument function as a one-argument function, or vice-versa. In this version 
A ( y ) is not assumed to be constant w.r.t  y, analogous to mpt2mptx 6193. (Contributed by AV, 30-Mar-2019.)
Hypothesis
Ref Expression
mpt2mptx2.1  |-  ( z  =  <. x ,  y
>.  ->  C  =  D )
Assertion
Ref Expression
mpt2mptx2  |-  ( z  e.  U_ y  e.  B  ( A  X.  { y } ) 
|->  C )  =  ( x  e.  A , 
y  e.  B  |->  D )
Distinct variable groups:    x, y,
z    x, A, z    x, B, z    x, C, y   
z, D
Allowed substitution hints:    A( y)    B( y)    C( z)    D( x, y)

Proof of Theorem mpt2mptx2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 df-mpt 4364 . 2  |-  ( z  e.  U_ y  e.  B  ( A  X.  { y } ) 
|->  C )  =  { <. z ,  w >.  |  ( z  e.  U_ y  e.  B  ( A  X.  { y } )  /\  w  =  C ) }
2 df-mpt2 6108 . . 3  |-  ( x  e.  A ,  y  e.  B  |->  D )  =  { <. <. x ,  y >. ,  w >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) }
3 eliunxp2 30733 . . . . . . 7  |-  ( z  e.  U_ y  e.  B  ( A  X.  { y } )  <->  E. x E. y ( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
) )
43anbi1i 695 . . . . . 6  |-  ( ( z  e.  U_ y  e.  B  ( A  X.  { y } )  /\  w  =  C )  <->  ( E. x E. y ( z  = 
<. x ,  y >.  /\  ( x  e.  A  /\  y  e.  B
) )  /\  w  =  C ) )
5 19.41vv 1921 . . . . . 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 649 . . . . . . . 8  |-  ( ( ( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  /\  w  =  C )  <->  ( z  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  C ) ) )
7 mpt2mptx2.1 . . . . . . . . . . 11  |-  ( z  =  <. x ,  y
>.  ->  C  =  D )
87eqeq2d 2454 . . . . . . . . . 10  |-  ( z  =  <. x ,  y
>.  ->  ( w  =  C  <->  w  =  D
) )
98anbi2d 703 . . . . . . . . 9  |-  ( z  =  <. x ,  y
>.  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  w  =  C
)  <->  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) ) )
109pm5.32i 637 . . . . . . . 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 249 . . . . . . 7  |-  ( ( ( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  /\  w  =  C )  <->  ( z  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) ) )
12112exbii 1635 . . . . . 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 273 . . . . 5  |-  ( ( z  e.  U_ y  e.  B  ( A  X.  { y } )  /\  w  =  C )  <->  E. x E. y
( z  =  <. x ,  y >.  /\  (
( x  e.  A  /\  y  e.  B
)  /\  w  =  D ) ) )
1413opabbii 4368 . . . 4  |-  { <. z ,  w >.  |  ( z  e.  U_ y  e.  B  ( A  X.  { y } )  /\  w  =  C ) }  =  { <. z ,  w >.  |  E. x E. y
( z  =  <. x ,  y >.  /\  (
( x  e.  A  /\  y  e.  B
)  /\  w  =  D ) ) }
15 dfoprab2 6145 . . . 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 2466 . . 3  |-  { <. z ,  w >.  |  ( z  e.  U_ y  e.  B  ( A  X.  { y } )  /\  w  =  C ) }  =  { <. <. x ,  y
>. ,  w >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) }
172, 16eqtr4i 2466 . 2  |-  ( x  e.  A ,  y  e.  B  |->  D )  =  { <. z ,  w >.  |  (
z  e.  U_ y  e.  B  ( A  X.  { y } )  /\  w  =  C ) }
181, 17eqtr4i 2466 1  |-  ( z  e.  U_ y  e.  B  ( A  X.  { y } ) 
|->  C )  =  ( x  e.  A , 
y  e.  B  |->  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   {csn 3889   <.cop 3895   U_ciun 4183   {copab 4361    e. cmpt 4362    X. cxp 4850   {coprab 6104    e. cmpt2 6105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-iun 4185  df-opab 4363  df-mpt 4364  df-xp 4858  df-rel 4859  df-oprab 6107  df-mpt2 6108
This theorem is referenced by:  dmmpt2ssx2  30739
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