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Theorem f2ndres 6142
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 2791 . . . . . . . 8  |-  y  e. 
_V
2 vex 2791 . . . . . . . 8  |-  z  e. 
_V
31, 2op2nda 5157 . . . . . . 7  |-  U. ran  {
<. y ,  z >. }  =  z
43eleq1i 2346 . . . . . 6  |-  ( U. ran  { <. y ,  z
>. }  e.  B  <->  z  e.  B )
54biimpri 197 . . . . 5  |-  ( z  e.  B  ->  U. ran  {
<. y ,  z >. }  e.  B )
65adantl 452 . . . 4  |-  ( ( y  e.  A  /\  z  e.  B )  ->  U. ran  { <. y ,  z >. }  e.  B )
76rgen2 2639 . . 3  |-  A. y  e.  A  A. z  e.  B  U. ran  { <. y ,  z >. }  e.  B
8 sneq 3651 . . . . . . 7  |-  ( x  =  <. y ,  z
>.  ->  { x }  =  { <. y ,  z
>. } )
98rneqd 4906 . . . . . 6  |-  ( x  =  <. y ,  z
>.  ->  ran  { x }  =  ran  { <. y ,  z >. } )
109unieqd 3838 . . . . 5  |-  ( x  =  <. y ,  z
>.  ->  U. ran  { x }  =  U. ran  { <. y ,  z >. } )
1110eleq1d 2349 . . . 4  |-  ( x  =  <. y ,  z
>.  ->  ( U. ran  { x }  e.  B  <->  U.
ran  { <. y ,  z
>. }  e.  B ) )
1211ralxp 4827 . . 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 200 . 2  |-  A. x  e.  ( A  X.  B
) U. ran  {
x }  e.  B
14 df-2nd 6123 . . . . 5  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
1514reseq1i 4951 . . . 4  |-  ( 2nd  |`  ( A  X.  B
) )  =  ( ( x  e.  _V  |->  U.
ran  { x } )  |`  ( A  X.  B
) )
16 ssv 3198 . . . . 5  |-  ( A  X.  B )  C_  _V
17 resmpt 5000 . . . . 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 8 . . . 4  |-  ( ( x  e.  _V  |->  U.
ran  { x } )  |`  ( A  X.  B
) )  =  ( x  e.  ( A  X.  B )  |->  U.
ran  { x } )
1915, 18eqtri 2303 . . 3  |-  ( 2nd  |`  ( A  X.  B
) )  =  ( x  e.  ( A  X.  B )  |->  U.
ran  { x } )
2019fmpt 5681 . 2  |-  ( A. x  e.  ( A  X.  B ) U. ran  { x }  e.  B  <->  ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B
) --> B )
2113, 20mpbi 199 1  |-  ( 2nd  |`  ( A  X.  B
) ) : ( A  X.  B ) --> B
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   {csn 3640   <.cop 3643   U.cuni 3827    e. cmpt 4077    X. cxp 4687   ran crn 4690    |` cres 4691   -->wf 5251   2ndc2nd 6121
This theorem is referenced by:  fo2ndres  6144  2ndcof  6148  fparlem2  6219  eucalgcvga  12756  2ndfcl  13972  gaid  14753  tx2cn  17304  txkgen  17346  xpinpreima  23290  xpinpreima2  23291  2ndmbfm  23566  filnetlem4  26330  hausgraph  27531
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-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-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-2nd 6123
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