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Theorem mpt2mptsx 6187
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 2791 . . . . . 6  |-  u  e. 
_V
2 vex 2791 . . . . . 6  |-  v  e. 
_V
31, 2op1std 6130 . . . . 5  |-  ( z  =  <. u ,  v
>.  ->  ( 1st `  z
)  =  u )
43csbeq1d 3087 . . . 4  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
51, 2op2ndd 6131 . . . . . 6  |-  ( z  =  <. u ,  v
>.  ->  ( 2nd `  z
)  =  v )
65csbeq1d 3087 . . . . 5  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 2nd `  z
)  /  y ]_ C  =  [_ v  / 
y ]_ C )
76csbeq2dv 3106 . . . 4  |-  ( z  =  <. u ,  v
>.  ->  [_ u  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
84, 7eqtrd 2315 . . 3  |-  ( z  =  <. u ,  v
>.  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
98mpt2mptx 5938 . 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 2419 . . . 4  |-  F/_ u
( { x }  X.  B )
11 nfcv 2419 . . . . 5  |-  F/_ x { u }
12 nfcsb1v 3113 . . . . 5  |-  F/_ x [_ u  /  x ]_ B
1311, 12nfxp 4715 . . . 4  |-  F/_ x
( { u }  X.  [_ u  /  x ]_ B )
14 sneq 3651 . . . . 5  |-  ( x  =  u  ->  { x }  =  { u } )
15 csbeq1a 3089 . . . . 5  |-  ( x  =  u  ->  B  =  [_ u  /  x ]_ B )
1614, 15xpeq12d 4714 . . . 4  |-  ( x  =  u  ->  ( { x }  X.  B )  =  ( { u }  X.  [_ u  /  x ]_ B ) )
1710, 13, 16cbviun 3939 . . 3  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  U_ u  e.  A  ( {
u }  X.  [_ u  /  x ]_ B
)
18 mpteq1 4100 . . 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 8 . 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 2419 . . 3  |-  F/_ u B
21 nfcv 2419 . . 3  |-  F/_ u C
22 nfcv 2419 . . 3  |-  F/_ v C
23 nfcsb1v 3113 . . 3  |-  F/_ x [_ u  /  x ]_ [_ v  /  y ]_ C
24 nfcv 2419 . . . 4  |-  F/_ y
u
25 nfcsb1v 3113 . . . 4  |-  F/_ y [_ v  /  y ]_ C
2624, 25nfcsb 3115 . . 3  |-  F/_ y [_ u  /  x ]_ [_ v  /  y ]_ C
27 csbeq1a 3089 . . . 4  |-  ( y  =  v  ->  C  =  [_ v  /  y ]_ C )
28 csbeq1a 3089 . . . 4  |-  ( x  =  u  ->  [_ v  /  y ]_ C  =  [_ u  /  x ]_ [_ v  /  y ]_ C )
2927, 28sylan9eqr 2337 . . 3  |-  ( ( x  =  u  /\  y  =  v )  ->  C  =  [_ u  /  x ]_ [_ v  /  y ]_ C
)
3020, 12, 21, 22, 23, 26, 15, 29cbvmpt2x 5924 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( u  e.  A ,  v  e. 
[_ u  /  x ]_ B  |->  [_ u  /  x ]_ [_ v  /  y ]_ C
)
319, 19, 303eqtr4ri 2314 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 1623   [_csb 3081   {csn 3640   <.cop 3643   U_ciun 3905    e. cmpt 4077    X. cxp 4687   ` cfv 5255    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121
This theorem is referenced by:  mpt2mpts  6188  ovmptss  6200
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-eu 2147  df-mo 2148  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-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123
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