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Theorem dfmpo 6171
Description: Alternate definition for the maps-to notation df-mpo 5830 (although it requires that  C be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
dfmpo.1  |-  C  e. 
_V
Assertion
Ref Expression
dfmpo  |-  ( 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 dfmpo
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 mpompts 6147 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( w  e.  ( A  X.  B
)  |->  [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C )
2 vex 2715 . . . . 5  |-  w  e. 
_V
3 1stexg 6116 . . . . 5  |-  ( w  e.  _V  ->  ( 1st `  w )  e. 
_V )
42, 3ax-mp 5 . . . 4  |-  ( 1st `  w )  e.  _V
5 2ndexg 6117 . . . . . 6  |-  ( w  e.  _V  ->  ( 2nd `  w )  e. 
_V )
62, 5ax-mp 5 . . . . 5  |-  ( 2nd `  w )  e.  _V
7 dfmpo.1 . . . . 5  |-  C  e. 
_V
86, 7csbexa 4094 . . . 4  |-  [_ ( 2nd `  w )  / 
y ]_ C  e.  _V
94, 8csbexa 4094 . . 3  |-  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  /  y ]_ C  e.  _V
109dfmpt 5645 . 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 2299 . . . . 5  |-  F/_ x w
12 nfcsb1v 3064 . . . . 5  |-  F/_ x [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C
1311, 12nfop 3758 . . . 4  |-  F/_ x <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  /  y ]_ C >.
1413nfsn 3620 . . 3  |-  F/_ x { <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  / 
y ]_ C >. }
15 nfcv 2299 . . . . 5  |-  F/_ y
w
16 nfcv 2299 . . . . . 6  |-  F/_ y
( 1st `  w
)
17 nfcsb1v 3064 . . . . . 6  |-  F/_ y [_ ( 2nd `  w
)  /  y ]_ C
1816, 17nfcsb 3068 . . . . 5  |-  F/_ y [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C
1915, 18nfop 3758 . . . 4  |-  F/_ y <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  /  y ]_ C >.
2019nfsn 3620 . . 3  |-  F/_ y { <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  / 
y ]_ C >. }
21 nfcv 2299 . . 3  |-  F/_ w { <. <. x ,  y
>. ,  C >. }
22 id 19 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  w  =  <. x ,  y >. )
23 csbopeq1a 6137 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C  =  C )
2422, 23opeq12d 3750 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  <. w ,  [_ ( 1st `  w )  /  x ]_ [_ ( 2nd `  w )  / 
y ]_ C >.  =  <. <.
x ,  y >. ,  C >. )
2524sneqd 3573 . . 3  |-  ( w  =  <. x ,  y
>.  ->  { <. w ,  [_ ( 1st `  w
)  /  x ]_ [_ ( 2nd `  w
)  /  y ]_ C >. }  =  { <. <. x ,  y
>. ,  C >. } )
2614, 20, 21, 25iunxpf 4735 . 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 2182 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 1335    e. wcel 2128   _Vcvv 2712   [_csb 3031   {csn 3560   <.cop 3563   U_ciun 3850    |-> cmpt 4026    X. cxp 4585   ` cfv 5171    e. cmpo 5827   1stc1st 6087   2ndc2nd 6088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170  ax-un 4394
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090
This theorem is referenced by: (None)
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