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Theorem mpt2mptx 5938
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 4079 . 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 5863 . . 3  |-  ( x  e.  A ,  y  e.  B  |->  D )  =  { <. <. x ,  y >. ,  w >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) }
3 eliunxp 4823 . . . . . . 7  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  <->  E. x E. y ( z  = 
<. x ,  y >.  /\  ( x  e.  A  /\  y  e.  B
) ) )
43anbi1i 676 . . . . . 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 1843 . . . . . 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 630 . . . . . . . 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 2294 . . . . . . . . . 10  |-  ( z  =  <. x ,  y
>.  ->  ( w  =  C  <->  w  =  D
) )
98anbi2d 684 . . . . . . . . 9  |-  ( z  =  <. x ,  y
>.  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  w  =  C
)  <->  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) ) )
109pm5.32i 618 . . . . . . . 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 240 . . . . . . 7  |-  ( ( ( z  =  <. x ,  y >.  /\  (
x  e.  A  /\  y  e.  B )
)  /\  w  =  C )  <->  ( z  =  <. x ,  y
>.  /\  ( ( x  e.  A  /\  y  e.  B )  /\  w  =  D ) ) )
12112exbii 1570 . . . . . 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 264 . . . . 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 4083 . . . 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 5895 . . . 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 2306 . . 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 2306 . 2  |-  ( x  e.  A ,  y  e.  B  |->  D )  =  { <. z ,  w >.  |  (
z  e.  U_ x  e.  A  ( {
x }  X.  B
)  /\  w  =  C ) }
181, 17eqtr4i 2306 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 358   E.wex 1528    = wceq 1623    e. wcel 1684   {csn 3640   <.cop 3643   U_ciun 3905   {copab 4076    e. cmpt 4077    X. cxp 4687   {coprab 5859    e. cmpt2 5860
This theorem is referenced by:  mpt2mpt  5939  mpt2mptsx  6187  dmmpt2ssx  6189  fmpt2x  6190  gsumcom2  15226
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-iun 3907  df-opab 4078  df-mpt 4079  df-xp 4695  df-rel 4696  df-oprab 5862  df-mpt2 5863
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