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Theorem f2ndres 5913
Description: Mapping of a restriction of the  2nd (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
f2ndres  |-  ( 2nd  |`  ( A  X.  B
) ) : ( A  X.  B ) --> B

Proof of Theorem f2ndres
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2622 . . . . . . . 8  |-  y  e. 
_V
2 vex 2622 . . . . . . . 8  |-  z  e. 
_V
31, 2op2nda 4902 . . . . . . 7  |-  U. ran  {
<. y ,  z >. }  =  z
43eleq1i 2153 . . . . . 6  |-  ( U. ran  { <. y ,  z
>. }  e.  B  <->  z  e.  B )
54biimpri 131 . . . . 5  |-  ( z  e.  B  ->  U. ran  {
<. y ,  z >. }  e.  B )
65adantl 271 . . . 4  |-  ( ( y  e.  A  /\  z  e.  B )  ->  U. ran  { <. y ,  z >. }  e.  B )
76rgen2 2459 . . 3  |-  A. y  e.  A  A. z  e.  B  U. ran  { <. y ,  z >. }  e.  B
8 sneq 3452 . . . . . . 7  |-  ( x  =  <. y ,  z
>.  ->  { x }  =  { <. y ,  z
>. } )
98rneqd 4652 . . . . . 6  |-  ( x  =  <. y ,  z
>.  ->  ran  { x }  =  ran  { <. y ,  z >. } )
109unieqd 3659 . . . . 5  |-  ( x  =  <. y ,  z
>.  ->  U. ran  { x }  =  U. ran  { <. y ,  z >. } )
1110eleq1d 2156 . . . 4  |-  ( x  =  <. y ,  z
>.  ->  ( U. ran  { x }  e.  B  <->  U.
ran  { <. y ,  z
>. }  e.  B ) )
1211ralxp 4567 . . 3  |-  ( A. x  e.  ( A  X.  B ) U. ran  { x }  e.  B  <->  A. y  e.  A  A. z  e.  B  U. ran  { <. y ,  z
>. }  e.  B )
137, 12mpbir 144 . 2  |-  A. x  e.  ( A  X.  B
) U. ran  {
x }  e.  B
14 df-2nd 5894 . . . . 5  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
1514reseq1i 4697 . . . 4  |-  ( 2nd  |`  ( A  X.  B
) )  =  ( ( x  e.  _V  |->  U.
ran  { x } )  |`  ( A  X.  B
) )
16 ssv 3044 . . . . 5  |-  ( A  X.  B )  C_  _V
17 resmpt 4747 . . . . 5  |-  ( ( A  X.  B ) 
C_  _V  ->  ( ( x  e.  _V  |->  U.
ran  { x } )  |`  ( A  X.  B
) )  =  ( x  e.  ( A  X.  B )  |->  U.
ran  { x } ) )
1816, 17ax-mp 7 . . . 4  |-  ( ( x  e.  _V  |->  U.
ran  { x } )  |`  ( A  X.  B
) )  =  ( x  e.  ( A  X.  B )  |->  U.
ran  { x } )
1915, 18eqtri 2108 . . 3  |-  ( 2nd  |`  ( A  X.  B
) )  =  ( x  e.  ( A  X.  B )  |->  U.
ran  { x } )
2019fmpt 5433 . 2  |-  ( A. x  e.  ( A  X.  B ) U. ran  { x }  e.  B  <->  ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B
) --> B )
2113, 20mpbi 143 1  |-  ( 2nd  |`  ( A  X.  B
) ) : ( A  X.  B ) --> B
Colors of variables: wff set class
Syntax hints:    = wceq 1289    e. wcel 1438   A.wral 2359   _Vcvv 2619    C_ wss 2997   {csn 3441   <.cop 3444   U.cuni 3648    |-> cmpt 3891    X. cxp 4426   ran crn 4429    |` cres 4430   -->wf 4998   2ndc2nd 5892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-2nd 5894
This theorem is referenced by:  fo2ndresm  5915  2ndcof  5917  f2ndf  5973  eucialgcvga  11122
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