ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfmpt2 Unicode version

Theorem dfmpt2 5970
Description: Alternate definition for the maps-to notation df-mpt2 5639 (although it requires that  C be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
dfmpt2.1  |-  C  e. 
_V
Assertion
Ref Expression
dfmpt2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  U_ x  e.  A  U_ y  e.  B  { <. <. x ,  y >. ,  C >. }
Distinct variable groups:    x, y, A   
x, B, y
Allowed substitution hints:    C( x, y)

Proof of Theorem dfmpt2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 mpt2mpts 5950 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( w  e.  ( A  X.  B
)  |->  [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C )
2 vex 2622 . . . . 5  |-  w  e. 
_V
3 1stexg 5920 . . . . 5  |-  ( w  e.  _V  ->  ( 1st `  w )  e. 
_V )
42, 3ax-mp 7 . . . 4  |-  ( 1st `  w )  e.  _V
5 2ndexg 5921 . . . . . 6  |-  ( w  e.  _V  ->  ( 2nd `  w )  e. 
_V )
62, 5ax-mp 7 . . . . 5  |-  ( 2nd `  w )  e.  _V
7 dfmpt2.1 . . . . 5  |-  C  e. 
_V
86, 7csbexa 3960 . . . 4  |-  [_ ( 2nd `  w )  / 
y ]_ C  e.  _V
94, 8csbexa 3960 . . 3  |-  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  /  y ]_ C  e.  _V
109dfmpt 5458 . 2  |-  ( w  e.  ( A  X.  B )  |->  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  /  y ]_ C )  =  U_ w  e.  ( A  X.  B ) { <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C >. }
11 nfcv 2228 . . . . 5  |-  F/_ x w
12 nfcsb1v 2961 . . . . 5  |-  F/_ x [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C
1311, 12nfop 3633 . . . 4  |-  F/_ x <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  /  y ]_ C >.
1413nfsn 3497 . . 3  |-  F/_ x { <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  / 
y ]_ C >. }
15 nfcv 2228 . . . . 5  |-  F/_ y
w
16 nfcv 2228 . . . . . 6  |-  F/_ y
( 1st `  w
)
17 nfcsb1v 2961 . . . . . 6  |-  F/_ y [_ ( 2nd `  w
)  /  y ]_ C
1816, 17nfcsb 2963 . . . . 5  |-  F/_ y [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C
1915, 18nfop 3633 . . . 4  |-  F/_ y <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  /  y ]_ C >.
2019nfsn 3497 . . 3  |-  F/_ y { <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  / 
y ]_ C >. }
21 nfcv 2228 . . 3  |-  F/_ w { <. <. x ,  y
>. ,  C >. }
22 id 19 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  w  =  <. x ,  y >. )
23 csbopeq1a 5940 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C  =  C )
2422, 23opeq12d 3625 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  / 
y ]_ C >.  =  <. <.
x ,  y >. ,  C >. )
2524sneqd 3454 . . 3  |-  ( w  =  <. x ,  y
>.  ->  { <. w ,  [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C >. }  =  { <. <. x ,  y
>. ,  C >. } )
2614, 20, 21, 25iunxpf 4572 . 2  |-  U_ w  e.  ( A  X.  B
) { <. w ,  [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C >. }  =  U_ x  e.  A  U_ y  e.  B  { <. <. x ,  y >. ,  C >. }
271, 10, 263eqtri 2112 1  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  U_ x  e.  A  U_ y  e.  B  { <. <. x ,  y >. ,  C >. }
Colors of variables: wff set class
Syntax hints:    = wceq 1289    e. wcel 1438   _Vcvv 2619   [_csb 2931   {csn 3441   <.cop 3444   U_ciun 3725    |-> cmpt 3891    X. cxp 4426   ` cfv 5002    |-> cmpt2 5636   1stc1st 5891   2ndc2nd 5892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator