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Theorem mpt2mpts 6204
Description: Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)
Assertion
Ref Expression
mpt2mpts  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e.  ( A  X.  B
)  |->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C )
Distinct variable groups:    x, y,
z, A    y, B, z    z, C    x, B
Allowed substitution hints:    C( x, y)

Proof of Theorem mpt2mpts
StepHypRef Expression
1 mpt2mptsx 6203 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e. 
U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
2 iunxpconst 4762 . . 3  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  ( A  X.  B )
3 mpteq1 4116 . . 3  |-  ( U_ x  e.  A  ( { x }  X.  B )  =  ( A  X.  B )  ->  ( z  e. 
U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )  =  ( z  e.  ( A  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ C ) )
42, 3ax-mp 8 . 2  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ C )  =  ( z  e.  ( A  X.  B ) 
|->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C )
51, 4eqtri 2316 1  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e.  ( A  X.  B
)  |->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C )
Colors of variables: wff set class
Syntax hints:    = wceq 1632   [_csb 3094   {csn 3653   U_ciun 3921    e. cmpt 4093    X. cxp 4703   ` cfv 5271    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  dfmpt2  6225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139
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