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Theorem f2ndres 5815
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 2577 . . . . . . . 8  |-  y  e. 
_V
2 vex 2577 . . . . . . . 8  |-  z  e. 
_V
31, 2op2nda 4833 . . . . . . 7  |-  U. ran  {
<. y ,  z >. }  =  z
43eleq1i 2119 . . . . . 6  |-  ( U. ran  { <. y ,  z
>. }  e.  B  <->  z  e.  B )
54biimpri 128 . . . . 5  |-  ( z  e.  B  ->  U. ran  {
<. y ,  z >. }  e.  B )
65adantl 266 . . . 4  |-  ( ( y  e.  A  /\  z  e.  B )  ->  U. ran  { <. y ,  z >. }  e.  B )
76rgen2 2422 . . 3  |-  A. y  e.  A  A. z  e.  B  U. ran  { <. y ,  z >. }  e.  B
8 sneq 3414 . . . . . . 7  |-  ( x  =  <. y ,  z
>.  ->  { x }  =  { <. y ,  z
>. } )
98rneqd 4591 . . . . . 6  |-  ( x  =  <. y ,  z
>.  ->  ran  { x }  =  ran  { <. y ,  z >. } )
109unieqd 3619 . . . . 5  |-  ( x  =  <. y ,  z
>.  ->  U. ran  { x }  =  U. ran  { <. y ,  z >. } )
1110eleq1d 2122 . . . 4  |-  ( x  =  <. y ,  z
>.  ->  ( U. ran  { x }  e.  B  <->  U.
ran  { <. y ,  z
>. }  e.  B ) )
1211ralxp 4507 . . 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 138 . 2  |-  A. x  e.  ( A  X.  B
) U. ran  {
x }  e.  B
14 df-2nd 5796 . . . . 5  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
1514reseq1i 4636 . . . 4  |-  ( 2nd  |`  ( A  X.  B
) )  =  ( ( x  e.  _V  |->  U.
ran  { x } )  |`  ( A  X.  B
) )
16 ssv 2993 . . . . 5  |-  ( A  X.  B )  C_  _V
17 resmpt 4684 . . . . 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 2076 . . 3  |-  ( 2nd  |`  ( A  X.  B
) )  =  ( x  e.  ( A  X.  B )  |->  U.
ran  { x } )
2019fmpt 5347 . 2  |-  ( A. x  e.  ( A  X.  B ) U. ran  { x }  e.  B  <->  ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B
) --> B )
2113, 20mpbi 137 1  |-  ( 2nd  |`  ( A  X.  B
) ) : ( A  X.  B ) --> B
Colors of variables: wff set class
Syntax hints:    = wceq 1259    e. wcel 1409   A.wral 2323   _Vcvv 2574    C_ wss 2945   {csn 3403   <.cop 3406   U.cuni 3608    |-> cmpt 3846    X. cxp 4371   ran crn 4374    |` cres 4375   -->wf 4926   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-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
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-rab 2332  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-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fv 4938  df-2nd 5796
This theorem is referenced by:  fo2ndresm  5817  2ndcof  5819  f2ndf  5875
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