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Theorem 2ndvalg 5795
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 3958 . . 3  |-  ( A  e.  _V  ->  { A }  e.  _V )
2 rnexg 4619 . . 3  |-  ( { A }  e.  _V  ->  ran  { A }  e.  _V )
3 uniexg 4195 . . 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 3411 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
65rneqd 4585 . . . 4  |-  ( x  =  A  ->  ran  { x }  =  ran  { A } )
76unieqd 3614 . . 3  |-  ( x  =  A  ->  U. ran  { x }  =  U. ran  { A } )
8 df-2nd 5793 . . 3  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
97, 8fvmptg 5274 . 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 1285    e. wcel 1434   _Vcvv 2602   {csn 3400   U.cuni 3603   ran crn 4366   ` cfv 4926   2ndc2nd 5791
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-iota 4891  df-fun 4928  df-fv 4934  df-2nd 5793
This theorem is referenced by:  2nd0  5797  op2nd  5799  elxp6  5821
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