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Theorem dfmpt2 6209
Description: Alternate definition for the "maps to" notation df-mpt2 5863 (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 6188 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( w  e.  ( A  X.  B
)  |->  [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C )
2 fvex 5539 . . . 4  |-  ( 1st `  w )  e.  _V
3 fvex 5539 . . . . 5  |-  ( 2nd `  w )  e.  _V
4 dfmpt2.1 . . . . 5  |-  C  e. 
_V
53, 4csbex 3092 . . . 4  |-  [_ ( 2nd `  w )  / 
y ]_ C  e.  _V
62, 5csbex 3092 . . 3  |-  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  /  y ]_ C  e.  _V
76dfmpt 5701 . 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 >. }
8 nfcv 2419 . . . . 5  |-  F/_ x w
9 nfcsb1v 3113 . . . . 5  |-  F/_ x [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C
108, 9nfop 3812 . . . 4  |-  F/_ x <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  /  y ]_ C >.
1110nfsn 3691 . . 3  |-  F/_ x { <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  / 
y ]_ C >. }
12 nfcv 2419 . . . . 5  |-  F/_ y
w
13 nfcv 2419 . . . . . 6  |-  F/_ y
( 1st `  w
)
14 nfcsb1v 3113 . . . . . 6  |-  F/_ y [_ ( 2nd `  w
)  /  y ]_ C
1513, 14nfcsb 3115 . . . . 5  |-  F/_ y [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C
1612, 15nfop 3812 . . . 4  |-  F/_ y <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  /  y ]_ C >.
1716nfsn 3691 . . 3  |-  F/_ y { <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  / 
y ]_ C >. }
18 nfcv 2419 . . 3  |-  F/_ w { <. <. x ,  y
>. ,  C >. }
19 id 19 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  w  =  <. x ,  y >. )
20 csbopeq1a 6173 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C  =  C )
2119, 20opeq12d 3804 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  / 
y ]_ C >.  =  <. <.
x ,  y >. ,  C >. )
2221sneqd 3653 . . 3  |-  ( w  =  <. x ,  y
>.  ->  { <. w ,  [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C >. }  =  { <. <. x ,  y
>. ,  C >. } )
2311, 17, 18, 22iunxpf 4832 . 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 >. }
241, 7, 233eqtri 2307 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 1623    e. wcel 1684   _Vcvv 2788   [_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:  fpar  6222
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-reu 2550  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-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123
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