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Theorem f2ndres 6336
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 2927 . . . . . . . 8  |-  y  e. 
_V
2 vex 2927 . . . . . . . 8  |-  z  e. 
_V
31, 2op2nda 5321 . . . . . . 7  |-  U. ran  {
<. y ,  z >. }  =  z
43eleq1i 2475 . . . . . 6  |-  ( U. ran  { <. y ,  z
>. }  e.  B  <->  z  e.  B )
54biimpri 198 . . . . 5  |-  ( z  e.  B  ->  U. ran  {
<. y ,  z >. }  e.  B )
65adantl 453 . . . 4  |-  ( ( y  e.  A  /\  z  e.  B )  ->  U. ran  { <. y ,  z >. }  e.  B )
76rgen2 2770 . . 3  |-  A. y  e.  A  A. z  e.  B  U. ran  { <. y ,  z >. }  e.  B
8 sneq 3793 . . . . . . 7  |-  ( x  =  <. y ,  z
>.  ->  { x }  =  { <. y ,  z
>. } )
98rneqd 5064 . . . . . 6  |-  ( x  =  <. y ,  z
>.  ->  ran  { x }  =  ran  { <. y ,  z >. } )
109unieqd 3994 . . . . 5  |-  ( x  =  <. y ,  z
>.  ->  U. ran  { x }  =  U. ran  { <. y ,  z >. } )
1110eleq1d 2478 . . . 4  |-  ( x  =  <. y ,  z
>.  ->  ( U. ran  { x }  e.  B  <->  U.
ran  { <. y ,  z
>. }  e.  B ) )
1211ralxp 4983 . . 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 201 . 2  |-  A. x  e.  ( A  X.  B
) U. ran  {
x }  e.  B
14 df-2nd 6317 . . . . 5  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
1514reseq1i 5109 . . . 4  |-  ( 2nd  |`  ( A  X.  B
) )  =  ( ( x  e.  _V  |->  U.
ran  { x } )  |`  ( A  X.  B
) )
16 ssv 3336 . . . . 5  |-  ( A  X.  B )  C_  _V
17 resmpt 5158 . . . . 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 2432 . . 3  |-  ( 2nd  |`  ( A  X.  B
) )  =  ( x  e.  ( A  X.  B )  |->  U.
ran  { x } )
2019fmpt 5857 . 2  |-  ( A. x  e.  ( A  X.  B ) U. ran  { x }  e.  B  <->  ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B
) --> B )
2113, 20mpbi 200 1  |-  ( 2nd  |`  ( A  X.  B
) ) : ( A  X.  B ) --> B
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   A.wral 2674   _Vcvv 2924    C_ wss 3288   {csn 3782   <.cop 3785   U.cuni 3983    e. cmpt 4234    X. cxp 4843   ran crn 4846    |` cres 4847   -->wf 5417   2ndc2nd 6315
This theorem is referenced by:  fo2ndres  6338  2ndcof  6342  fparlem2  6414  f2ndf  6419  eucalgcvga  13040  2ndfcl  14258  gaid  15039  tx2cn  17603  txkgen  17645  xpinpreima  24265  xpinpreima2  24266  2ndmbfm  24572  filnetlem4  26308  hausgraph  27407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-2nd 6317
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