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Theorem dfmpt2 6897
Description: Alternate definition for the "maps to" notation df-mpt2 6310 (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 6871 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( w  e.  ( A  X.  B
)  |->  [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C )
2 dfmpt2.1 . . . . 5  |-  C  e. 
_V
32csbex 4559 . . . 4  |-  [_ ( 2nd `  w )  / 
y ]_ C  e.  _V
43csbex 4559 . . 3  |-  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  /  y ]_ C  e.  _V
54dfmpt 6084 . 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 >. }
6 nfcv 2580 . . . . 5  |-  F/_ x w
7 nfcsb1v 3411 . . . . 5  |-  F/_ x [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C
86, 7nfop 4203 . . . 4  |-  F/_ x <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  /  y ]_ C >.
98nfsn 4057 . . 3  |-  F/_ x { <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  / 
y ]_ C >. }
10 nfcv 2580 . . . . 5  |-  F/_ y
w
11 nfcv 2580 . . . . . 6  |-  F/_ y
( 1st `  w
)
12 nfcsb1v 3411 . . . . . 6  |-  F/_ y [_ ( 2nd `  w
)  /  y ]_ C
1311, 12nfcsb 3413 . . . . 5  |-  F/_ y [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C
1410, 13nfop 4203 . . . 4  |-  F/_ y <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  /  y ]_ C >.
1514nfsn 4057 . . 3  |-  F/_ y { <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  / 
y ]_ C >. }
16 nfcv 2580 . . 3  |-  F/_ w { <. <. x ,  y
>. ,  C >. }
17 id 22 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  w  =  <. x ,  y >. )
18 csbopeq1a 6860 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C  =  C )
1917, 18opeq12d 4195 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  / 
y ]_ C >.  =  <. <.
x ,  y >. ,  C >. )
2019sneqd 4010 . . 3  |-  ( w  =  <. x ,  y
>.  ->  { <. w ,  [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C >. }  =  { <. <. x ,  y
>. ,  C >. } )
219, 15, 16, 20iunxpf 5002 . 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 >. }
221, 5, 213eqtri 2455 1  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  U_ x  e.  A  U_ y  e.  B  { <. <. x ,  y >. ,  C >. }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    e. wcel 1872   _Vcvv 3080   [_csb 3395   {csn 3998   <.cop 4004   U_ciun 4299    |-> cmpt 4482    X. cxp 4851   ` cfv 5601    |-> cmpt2 6307   1stc1st 6805   2ndc2nd 6806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808
This theorem is referenced by:  fpar  6911
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