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Theorem f2ndres 6024
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 2661 . . . . . . . 8  |-  y  e. 
_V
2 vex 2661 . . . . . . . 8  |-  z  e. 
_V
31, 2op2nda 4991 . . . . . . 7  |-  U. ran  {
<. y ,  z >. }  =  z
43eleq1i 2181 . . . . . 6  |-  ( U. ran  { <. y ,  z
>. }  e.  B  <->  z  e.  B )
54biimpri 132 . . . . 5  |-  ( z  e.  B  ->  U. ran  {
<. y ,  z >. }  e.  B )
65adantl 273 . . . 4  |-  ( ( y  e.  A  /\  z  e.  B )  ->  U. ran  { <. y ,  z >. }  e.  B )
76rgen2 2493 . . 3  |-  A. y  e.  A  A. z  e.  B  U. ran  { <. y ,  z >. }  e.  B
8 sneq 3506 . . . . . . 7  |-  ( x  =  <. y ,  z
>.  ->  { x }  =  { <. y ,  z
>. } )
98rneqd 4736 . . . . . 6  |-  ( x  =  <. y ,  z
>.  ->  ran  { x }  =  ran  { <. y ,  z >. } )
109unieqd 3715 . . . . 5  |-  ( x  =  <. y ,  z
>.  ->  U. ran  { x }  =  U. ran  { <. y ,  z >. } )
1110eleq1d 2184 . . . 4  |-  ( x  =  <. y ,  z
>.  ->  ( U. ran  { x }  e.  B  <->  U.
ran  { <. y ,  z
>. }  e.  B ) )
1211ralxp 4650 . . 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 145 . 2  |-  A. x  e.  ( A  X.  B
) U. ran  {
x }  e.  B
14 df-2nd 6005 . . . . 5  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
1514reseq1i 4783 . . . 4  |-  ( 2nd  |`  ( A  X.  B
) )  =  ( ( x  e.  _V  |->  U.
ran  { x } )  |`  ( A  X.  B
) )
16 ssv 3087 . . . . 5  |-  ( A  X.  B )  C_  _V
17 resmpt 4835 . . . . 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 5 . . . 4  |-  ( ( x  e.  _V  |->  U.
ran  { x } )  |`  ( A  X.  B
) )  =  ( x  e.  ( A  X.  B )  |->  U.
ran  { x } )
1915, 18eqtri 2136 . . 3  |-  ( 2nd  |`  ( A  X.  B
) )  =  ( x  e.  ( A  X.  B )  |->  U.
ran  { x } )
2019fmpt 5536 . 2  |-  ( A. x  e.  ( A  X.  B ) U. ran  { x }  e.  B  <->  ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B
) --> B )
2113, 20mpbi 144 1  |-  ( 2nd  |`  ( A  X.  B
) ) : ( A  X.  B ) --> B
Colors of variables: wff set class
Syntax hints:    = wceq 1314    e. wcel 1463   A.wral 2391   _Vcvv 2658    C_ wss 3039   {csn 3495   <.cop 3498   U.cuni 3704    |-> cmpt 3957    X. cxp 4505   ran crn 4508    |` cres 4509   -->wf 5087   2ndc2nd 6003
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-2nd 6005
This theorem is referenced by:  fo2ndresm  6026  2ndcof  6028  f2ndf  6089  eucalgcvga  11635  tx2cn  12334
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