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Theorem mpt2mptsx 6417
Description: Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
mpt2mptsx  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e. 
U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
Distinct variable groups:    x, y,
z, A    y, B, z    z, C
Allowed substitution hints:    B( x)    C( x, y)

Proof of Theorem mpt2mptsx
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2961 . . . . . 6  |-  u  e. 
_V
2 vex 2961 . . . . . 6  |-  v  e. 
_V
31, 2op1std 6360 . . . . 5  |-  ( z  =  <. u ,  v
>.  ->  ( 1st `  z
)  =  u )
43csbeq1d 3259 . . . 4  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
51, 2op2ndd 6361 . . . . . 6  |-  ( z  =  <. u ,  v
>.  ->  ( 2nd `  z
)  =  v )
65csbeq1d 3259 . . . . 5  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 2nd `  z
)  /  y ]_ C  =  [_ v  / 
y ]_ C )
76csbeq2dv 3278 . . . 4  |-  ( z  =  <. u ,  v
>.  ->  [_ u  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
84, 7eqtrd 2470 . . 3  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
98mpt2mptx 6167 . 2  |-  ( z  e.  U_ u  e.  A  ( { u }  X.  [_ u  /  x ]_ B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ C )  =  ( u  e.  A ,  v  e.  [_ u  /  x ]_ B  |->  [_ u  /  x ]_ [_ v  /  y ]_ C
)
10 nfcv 2574 . . . 4  |-  F/_ u
( { x }  X.  B )
11 nfcv 2574 . . . . 5  |-  F/_ x { u }
12 nfcsb1v 3285 . . . . 5  |-  F/_ x [_ u  /  x ]_ B
1311, 12nfxp 4907 . . . 4  |-  F/_ x
( { u }  X.  [_ u  /  x ]_ B )
14 sneq 3827 . . . . 5  |-  ( x  =  u  ->  { x }  =  { u } )
15 csbeq1a 3261 . . . . 5  |-  ( x  =  u  ->  B  =  [_ u  /  x ]_ B )
1614, 15xpeq12d 4906 . . . 4  |-  ( x  =  u  ->  ( { x }  X.  B )  =  ( { u }  X.  [_ u  /  x ]_ B ) )
1710, 13, 16cbviun 4130 . . 3  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  U_ u  e.  A  ( {
u }  X.  [_ u  /  x ]_ B
)
18 mpteq1 4292 . . 3  |-  ( U_ x  e.  A  ( { x }  X.  B )  =  U_ u  e.  A  ( { u }  X.  [_ u  /  x ]_ B )  ->  (
z  e.  U_ x  e.  A  ( {
x }  X.  B
)  |->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C )  =  ( z  e.  U_ u  e.  A  ( {
u }  X.  [_ u  /  x ]_ B
)  |->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C ) )
1917, 18ax-mp 5 . 2  |-  ( z  e.  U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ C )  =  ( z  e.  U_ u  e.  A  ( { u }  X.  [_ u  /  x ]_ B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
20 nfcv 2574 . . 3  |-  F/_ u B
21 nfcv 2574 . . 3  |-  F/_ u C
22 nfcv 2574 . . 3  |-  F/_ v C
23 nfcsb1v 3285 . . 3  |-  F/_ x [_ u  /  x ]_ [_ v  /  y ]_ C
24 nfcv 2574 . . . 4  |-  F/_ y
u
25 nfcsb1v 3285 . . . 4  |-  F/_ y [_ v  /  y ]_ C
2624, 25nfcsb 3287 . . 3  |-  F/_ y [_ u  /  x ]_ [_ v  /  y ]_ C
27 csbeq1a 3261 . . . 4  |-  ( y  =  v  ->  C  =  [_ v  /  y ]_ C )
28 csbeq1a 3261 . . . 4  |-  ( x  =  u  ->  [_ v  /  y ]_ C  =  [_ u  /  x ]_ [_ v  /  y ]_ C )
2927, 28sylan9eqr 2492 . . 3  |-  ( ( x  =  u  /\  y  =  v )  ->  C  =  [_ u  /  x ]_ [_ v  /  y ]_ C
)
3020, 12, 21, 22, 23, 26, 15, 29cbvmpt2x 6153 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( u  e.  A ,  v  e. 
[_ u  /  x ]_ B  |->  [_ u  /  x ]_ [_ v  /  y ]_ C
)
319, 19, 303eqtr4ri 2469 1  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e. 
U_ x  e.  A  ( { x }  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
Colors of variables: wff set class
Syntax hints:    = wceq 1653   [_csb 3253   {csn 3816   <.cop 3819   U_ciun 4095    e. cmpt 4269    X. cxp 4879   ` cfv 5457    e. cmpt2 6086   1stc1st 6350   2ndc2nd 6351
This theorem is referenced by:  mpt2mpts  6418  ovmptss  6431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fv 5465  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353
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