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Theorem 2ndvalg 6007
Description: The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
2ndvalg  |-  ( A  e.  _V  ->  ( 2nd `  A )  = 
U. ran  { A } )

Proof of Theorem 2ndvalg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 snexg 4076 . . 3  |-  ( A  e.  _V  ->  { A }  e.  _V )
2 rnexg 4772 . . 3  |-  ( { A }  e.  _V  ->  ran  { A }  e.  _V )
3 uniexg 4329 . . 3  |-  ( ran 
{ A }  e.  _V  ->  U. ran  { A }  e.  _V )
41, 2, 33syl 17 . 2  |-  ( A  e.  _V  ->  U. ran  { A }  e.  _V )
5 sneq 3506 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
65rneqd 4736 . . . 4  |-  ( x  =  A  ->  ran  { x }  =  ran  { A } )
76unieqd 3715 . . 3  |-  ( x  =  A  ->  U. ran  { x }  =  U. ran  { A } )
8 df-2nd 6005 . . 3  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
97, 8fvmptg 5463 . 2  |-  ( ( A  e.  _V  /\  U.
ran  { A }  e.  _V )  ->  ( 2nd `  A )  =  U. ran  { A } )
104, 9mpdan 415 1  |-  ( A  e.  _V  ->  ( 2nd `  A )  = 
U. ran  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    e. wcel 1463   _Vcvv 2658   {csn 3495   U.cuni 3704   ran crn 4508   ` cfv 5091   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-13 1474  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  ax-un 4323
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-v 2660  df-sbc 2881  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-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-iota 5056  df-fun 5093  df-fv 5099  df-2nd 6005
This theorem is referenced by:  2nd0  6009  op2nd  6011  elxp6  6033
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