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Theorem mpt2mptxf 27673
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.) (Revised by Thierry Arnoux, 31-Mar-2018.)
Hypotheses
Ref Expression
mpt2mptxf.0  |-  F/_ x C
mpt2mptxf.1  |-  F/_ y C
mpt2mptxf.2  |-  ( z  =  <. x ,  y
>.  ->  C  =  D )
Assertion
Ref Expression
mpt2mptxf  |-  ( 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    z, D
Allowed substitution hints:    B( x)    C( x, y, z)    D( x, y)

Proof of Theorem mpt2mptxf
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 df-mpt 4517 . 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 6301 . . 3  |-  ( x  e.  A ,  y  e.  B  |->  D )  =  { <. <. x ,  y >. ,  w >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) }
3 eliunxp 5150 . . . . . . 7  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  <->  E. x E. y ( z  = 
<. x ,  y >.  /\  ( x  e.  A  /\  y  e.  B
) ) )
43anbi1i 695 . . . . . 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 mpt2mptxf.1 . . . . . . . . . 10  |-  F/_ y C
65nfeq2 2636 . . . . . . . . 9  |-  F/ y  w  =  C
7619.41 1972 . . . . . . . 8  |-  ( E. y ( ( z  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  e.  B ) )  /\  w  =  C )  <->  ( E. y ( z  =  <. x ,  y
>.  /\  ( x  e.  A  /\  y  e.  B ) )  /\  w  =  C )
)
87exbii 1668 . . . . . . 7  |-  ( 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 ) )
9 mpt2mptxf.0 . . . . . . . . 9  |-  F/_ x C
109nfeq2 2636 . . . . . . . 8  |-  F/ x  w  =  C
111019.41 1972 . . . . . . 7  |-  ( 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 )
)
128, 11bitri 249 . . . . . 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 )
)
13 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 ) ) )
14 mpt2mptxf.2 . . . . . . . . . . 11  |-  ( z  =  <. x ,  y
>.  ->  C  =  D )
1514eqeq2d 2471 . . . . . . . . . 10  |-  ( z  =  <. x ,  y
>.  ->  ( w  =  C  <->  w  =  D
) )
1615anbi2d 703 . . . . . . . . 9  |-  ( z  =  <. x ,  y
>.  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  w  =  C
)  <->  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) ) )
1716pm5.32i 637 . . . . . . . 8  |-  ( ( z  =  <. x ,  y >.  /\  (
( x  e.  A  /\  y  e.  B
)  /\  w  =  C ) )  <->  ( z  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) ) )
1813, 17bitri 249 . . . . . . 7  |-  ( ( ( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  /\  w  =  C )  <->  ( z  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) ) )
19182exbii 1669 . . . . . 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 ) ) )
204, 12, 193bitr2i 273 . . . . 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 ) ) )
2120opabbii 4521 . . . 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 ) ) }
22 dfoprab2 6342 . . . 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 ) ) }
2321, 22eqtr4i 2489 . . 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 ) }
242, 23eqtr4i 2489 . 2  |-  ( x  e.  A ,  y  e.  B  |->  D )  =  { <. z ,  w >.  |  (
z  e.  U_ x  e.  A  ( {
x }  X.  B
)  /\  w  =  C ) }
251, 24eqtr4i 2489 1  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395   E.wex 1613    e. wcel 1819   F/_wnfc 2605   {csn 4032   <.cop 4038   U_ciun 4332   {copab 4514    |-> cmpt 4515    X. cxp 5006   {coprab 6297    |-> cmpt2 6298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-iun 4334  df-opab 4516  df-mpt 4517  df-xp 5014  df-rel 5015  df-oprab 6300  df-mpt2 6301
This theorem is referenced by:  gsummpt2co  27931
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