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Theorem dfmpt2 5844
Description: Alternate definition for the "maps to" notation df-mpt2 5517 (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 5824 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( w  e.  ( A  X.  B
)  |->  [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C )
2 vex 2560 . . . . 5  |-  w  e. 
_V
3 1stexg 5794 . . . . 5  |-  ( w  e.  _V  ->  ( 1st `  w )  e. 
_V )
42, 3ax-mp 7 . . . 4  |-  ( 1st `  w )  e.  _V
5 2ndexg 5795 . . . . . 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 3886 . . . 4  |-  [_ ( 2nd `  w )  / 
y ]_ C  e.  _V
94, 8csbexa 3886 . . 3  |-  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  /  y ]_ C  e.  _V
109dfmpt 5340 . 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 2178 . . . . 5  |-  F/_ x w
12 nfcsb1v 2882 . . . . 5  |-  F/_ x [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C
1311, 12nfop 3565 . . . 4  |-  F/_ x <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  /  y ]_ C >.
1413nfsn 3430 . . 3  |-  F/_ x { <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  / 
y ]_ C >. }
15 nfcv 2178 . . . . 5  |-  F/_ y
w
16 nfcv 2178 . . . . . 6  |-  F/_ y
( 1st `  w
)
17 nfcsb1v 2882 . . . . . 6  |-  F/_ y [_ ( 2nd `  w
)  /  y ]_ C
1816, 17nfcsb 2884 . . . . 5  |-  F/_ y [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C
1915, 18nfop 3565 . . . 4  |-  F/_ y <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  /  y ]_ C >.
2019nfsn 3430 . . 3  |-  F/_ y { <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  / 
y ]_ C >. }
21 nfcv 2178 . . 3  |-  F/_ w { <. <. x ,  y
>. ,  C >. }
22 id 19 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  w  =  <. x ,  y >. )
23 csbopeq1a 5814 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C  =  C )
2422, 23opeq12d 3557 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  / 
y ]_ C >.  =  <. <.
x ,  y >. ,  C >. )
2524sneqd 3388 . . 3  |-  ( w  =  <. x ,  y
>.  ->  { <. w ,  [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C >. }  =  { <. <. x ,  y
>. ,  C >. } )
2614, 20, 21, 25iunxpf 4484 . 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 2064 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 1243    e. wcel 1393   _Vcvv 2557   [_csb 2852   {csn 3375   <.cop 3378   U_ciun 3657    |-> cmpt 3818    X. cxp 4343   ` cfv 4902    |-> cmpt2 5514   1stc1st 5765   2ndc2nd 5766
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768
This theorem is referenced by: (None)
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