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Theorem dfmpt2 5872
Description: Alternate definition for the "maps to" notation df-mpt2 5545 (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 5852 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( w  e.  ( A  X.  B
)  |->  [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C )
2 vex 2577 . . . . 5  |-  w  e. 
_V
3 1stexg 5822 . . . . 5  |-  ( w  e.  _V  ->  ( 1st `  w )  e. 
_V )
42, 3ax-mp 7 . . . 4  |-  ( 1st `  w )  e.  _V
5 2ndexg 5823 . . . . . 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 3914 . . . 4  |-  [_ ( 2nd `  w )  / 
y ]_ C  e.  _V
94, 8csbexa 3914 . . 3  |-  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  /  y ]_ C  e.  _V
109dfmpt 5368 . 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 2194 . . . . 5  |-  F/_ x w
12 nfcsb1v 2910 . . . . 5  |-  F/_ x [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C
1311, 12nfop 3593 . . . 4  |-  F/_ x <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  /  y ]_ C >.
1413nfsn 3458 . . 3  |-  F/_ x { <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  / 
y ]_ C >. }
15 nfcv 2194 . . . . 5  |-  F/_ y
w
16 nfcv 2194 . . . . . 6  |-  F/_ y
( 1st `  w
)
17 nfcsb1v 2910 . . . . . 6  |-  F/_ y [_ ( 2nd `  w
)  /  y ]_ C
1816, 17nfcsb 2912 . . . . 5  |-  F/_ y [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C
1915, 18nfop 3593 . . . 4  |-  F/_ y <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  /  y ]_ C >.
2019nfsn 3458 . . 3  |-  F/_ y { <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  / 
y ]_ C >. }
21 nfcv 2194 . . 3  |-  F/_ w { <. <. x ,  y
>. ,  C >. }
22 id 19 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  w  =  <. x ,  y >. )
23 csbopeq1a 5842 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C  =  C )
2422, 23opeq12d 3585 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  / 
y ]_ C >.  =  <. <.
x ,  y >. ,  C >. )
2524sneqd 3416 . . 3  |-  ( w  =  <. x ,  y
>.  ->  { <. w ,  [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C >. }  =  { <. <. x ,  y
>. ,  C >. } )
2614, 20, 21, 25iunxpf 4512 . 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 2080 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 1259    e. wcel 1409   _Vcvv 2574   [_csb 2880   {csn 3403   <.cop 3406   U_ciun 3685    |-> cmpt 3846    X. cxp 4371   ` cfv 4930    |-> cmpt2 5542   1stc1st 5793   2ndc2nd 5794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796
This theorem is referenced by: (None)
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