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Theorem 2ndvalg 5914
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 4019 . . 3  |-  ( A  e.  _V  ->  { A }  e.  _V )
2 rnexg 4698 . . 3  |-  ( { A }  e.  _V  ->  ran  { A }  e.  _V )
3 uniexg 4265 . . 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 3457 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
65rneqd 4664 . . . 4  |-  ( x  =  A  ->  ran  { x }  =  ran  { A } )
76unieqd 3664 . . 3  |-  ( x  =  A  ->  U. ran  { x }  =  U. ran  { A } )
8 df-2nd 5912 . . 3  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
97, 8fvmptg 5380 . 2  |-  ( ( A  e.  _V  /\  U.
ran  { A }  e.  _V )  ->  ( 2nd `  A )  =  U. ran  { A } )
104, 9mpdan 412 1  |-  ( A  e.  _V  ->  ( 2nd `  A )  = 
U. ran  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   _Vcvv 2619   {csn 3446   U.cuni 3653   ran crn 4439   ` cfv 5015   2ndc2nd 5910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fv 5023  df-2nd 5912
This theorem is referenced by:  2nd0  5916  op2nd  5918  elxp6  5940
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