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Theorem 2ndval 6141
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
2ndval  |-  ( 2nd `  A )  =  U. ran  { A }

Proof of Theorem 2ndval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sneq 3664 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
21rneqd 4922 . . . 4  |-  ( x  =  A  ->  ran  { x }  =  ran  { A } )
32unieqd 3854 . . 3  |-  ( x  =  A  ->  U. ran  { x }  =  U. ran  { A } )
4 df-2nd 6139 . . 3  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
5 snex 4232 . . . . 5  |-  { A }  e.  _V
65rnex 4958 . . . 4  |-  ran  { A }  e.  _V
76uniex 4532 . . 3  |-  U. ran  { A }  e.  _V
83, 4, 7fvmpt 5618 . 2  |-  ( A  e.  _V  ->  ( 2nd `  A )  = 
U. ran  { A } )
9 fvprc 5535 . . 3  |-  ( -.  A  e.  _V  ->  ( 2nd `  A )  =  (/) )
10 snprc 3708 . . . . . . . 8  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1110biimpi 186 . . . . . . 7  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1211rneqd 4922 . . . . . 6  |-  ( -.  A  e.  _V  ->  ran 
{ A }  =  ran  (/) )
13 rn0 4952 . . . . . 6  |-  ran  (/)  =  (/)
1412, 13syl6eq 2344 . . . . 5  |-  ( -.  A  e.  _V  ->  ran 
{ A }  =  (/) )
1514unieqd 3854 . . . 4  |-  ( -.  A  e.  _V  ->  U.
ran  { A }  =  U. (/) )
16 uni0 3870 . . . 4  |-  U. (/)  =  (/)
1715, 16syl6eq 2344 . . 3  |-  ( -.  A  e.  _V  ->  U.
ran  { A }  =  (/) )
189, 17eqtr4d 2331 . 2  |-  ( -.  A  e.  _V  ->  ( 2nd `  A )  =  U. ran  { A } )
198, 18pm2.61i 156 1  |-  ( 2nd `  A )  =  U. ran  { A }
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   {csn 3653   U.cuni 3843   ran crn 4706   ` cfv 5271   2ndc2nd 6137
This theorem is referenced by:  2nd0  6143  op2nd  6145  2nd2val  6162  elxp6  6167  2ndnpr  23261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-2nd 6139
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