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Theorem mpt2mptx2 40389
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 6414. (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 4477 . 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 6320 . . 3  |-  ( x  e.  A ,  y  e.  B  |->  D )  =  { <. <. x ,  y >. ,  w >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) }
3 eliunxp2 40388 . . . . . . 7  |-  ( z  e.  U_ y  e.  B  ( A  X.  { y } )  <->  E. x E. y ( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
) )
43anbi1i 706 . . . . . 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 1842 . . . . . 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 659 . . . . . . . 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 2472 . . . . . . . . . 10  |-  ( z  =  <. x ,  y
>.  ->  ( w  =  C  <->  w  =  D
) )
98anbi2d 715 . . . . . . . . 9  |-  ( z  =  <. x ,  y
>.  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  w  =  C
)  <->  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) ) )
109pm5.32i 647 . . . . . . . 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 257 . . . . . . 7  |-  ( ( ( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  /\  w  =  C )  <->  ( z  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) ) )
12112exbii 1730 . . . . . 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 281 . . . . 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 4481 . . . 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 6364 . . . 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 2487 . . 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 2487 . 2  |-  ( x  e.  A ,  y  e.  B  |->  D )  =  { <. z ,  w >.  |  (
z  e.  U_ y  e.  B  ( A  X.  { y } )  /\  w  =  C ) }
181, 17eqtr4i 2487 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 375    = wceq 1455   E.wex 1674    e. wcel 1898   {csn 3980   <.cop 3986   U_ciun 4292   {copab 4474    |-> cmpt 4475    X. cxp 4851   {coprab 6316    |-> cmpt2 6317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-iun 4294  df-opab 4476  df-mpt 4477  df-xp 4859  df-rel 4860  df-oprab 6319  df-mpt2 6320
This theorem is referenced by:  dmmpt2ssx2  40391
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