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Theorem mpt2mptsx 6857
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 3121 . . . . . 6  |-  u  e. 
_V
2 vex 3121 . . . . . 6  |-  v  e. 
_V
31, 2op1std 6804 . . . . 5  |-  ( z  =  <. u ,  v
>.  ->  ( 1st `  z
)  =  u )
43csbeq1d 3447 . . . 4  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
51, 2op2ndd 6805 . . . . . 6  |-  ( z  =  <. u ,  v
>.  ->  ( 2nd `  z
)  =  v )
65csbeq1d 3447 . . . . 5  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 2nd `  z
)  /  y ]_ C  =  [_ v  / 
y ]_ C )
76csbeq2dv 3840 . . . 4  |-  ( z  =  <. u ,  v
>.  ->  [_ u  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
84, 7eqtrd 2508 . . 3  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
98mpt2mptx 6387 . 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 2629 . . . 4  |-  F/_ u
( { x }  X.  B )
11 nfcv 2629 . . . . 5  |-  F/_ x { u }
12 nfcsb1v 3456 . . . . 5  |-  F/_ x [_ u  /  x ]_ B
1311, 12nfxp 5031 . . . 4  |-  F/_ x
( { u }  X.  [_ u  /  x ]_ B )
14 sneq 4042 . . . . 5  |-  ( x  =  u  ->  { x }  =  { u } )
15 csbeq1a 3449 . . . . 5  |-  ( x  =  u  ->  B  =  [_ u  /  x ]_ B )
1614, 15xpeq12d 5029 . . . 4  |-  ( x  =  u  ->  ( { x }  X.  B )  =  ( { u }  X.  [_ u  /  x ]_ B ) )
1710, 13, 16cbviun 4367 . . 3  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  U_ u  e.  A  ( {
u }  X.  [_ u  /  x ]_ B
)
18 mpteq1 4532 . . 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 2629 . . 3  |-  F/_ u B
21 nfcv 2629 . . 3  |-  F/_ u C
22 nfcv 2629 . . 3  |-  F/_ v C
23 nfcsb1v 3456 . . 3  |-  F/_ x [_ u  /  x ]_ [_ v  /  y ]_ C
24 nfcv 2629 . . . 4  |-  F/_ y
u
25 nfcsb1v 3456 . . . 4  |-  F/_ y [_ v  /  y ]_ C
2624, 25nfcsb 3458 . . 3  |-  F/_ y [_ u  /  x ]_ [_ v  /  y ]_ C
27 csbeq1a 3449 . . . 4  |-  ( y  =  v  ->  C  =  [_ v  /  y ]_ C )
28 csbeq1a 3449 . . . 4  |-  ( x  =  u  ->  [_ v  /  y ]_ C  =  [_ u  /  x ]_ [_ v  /  y ]_ C )
2927, 28sylan9eqr 2530 . . 3  |-  ( ( x  =  u  /\  y  =  v )  ->  C  =  [_ u  /  x ]_ [_ v  /  y ]_ C
)
3020, 12, 21, 22, 23, 26, 15, 29cbvmpt2x 6369 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( u  e.  A ,  v  e. 
[_ u  /  x ]_ B  |->  [_ u  /  x ]_ [_ v  /  y ]_ C
)
319, 19, 303eqtr4ri 2507 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 setvar class
Syntax hints:    = wceq 1379   [_csb 3440   {csn 4032   <.cop 4038   U_ciun 4330    |-> cmpt 4510    X. cxp 5002   ` cfv 5593    |-> cmpt2 6296   1stc1st 6792   2ndc2nd 6793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fv 5601  df-oprab 6298  df-mpt2 6299  df-1st 6794  df-2nd 6795
This theorem is referenced by:  mpt2mpts  6858  ovmptss  6874
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