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Theorem 2ndval 6125
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 3651 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
21rneqd 4906 . . . 4  |-  ( x  =  A  ->  ran  { x }  =  ran  { A } )
32unieqd 3838 . . 3  |-  ( x  =  A  ->  U. ran  { x }  =  U. ran  { A } )
4 df-2nd 6123 . . 3  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
5 snex 4216 . . . . 5  |-  { A }  e.  _V
65rnex 4942 . . . 4  |-  ran  { A }  e.  _V
76uniex 4516 . . 3  |-  U. ran  { A }  e.  _V
83, 4, 7fvmpt 5602 . 2  |-  ( A  e.  _V  ->  ( 2nd `  A )  = 
U. ran  { A } )
9 fvprc 5519 . . 3  |-  ( -.  A  e.  _V  ->  ( 2nd `  A )  =  (/) )
10 snprc 3695 . . . . . . . 8  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1110biimpi 186 . . . . . . 7  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1211rneqd 4906 . . . . . 6  |-  ( -.  A  e.  _V  ->  ran 
{ A }  =  ran  (/) )
13 rn0 4936 . . . . . 6  |-  ran  (/)  =  (/)
1412, 13syl6eq 2331 . . . . 5  |-  ( -.  A  e.  _V  ->  ran 
{ A }  =  (/) )
1514unieqd 3838 . . . 4  |-  ( -.  A  e.  _V  ->  U.
ran  { A }  =  U. (/) )
16 uni0 3854 . . . 4  |-  U. (/)  =  (/)
1715, 16syl6eq 2331 . . 3  |-  ( -.  A  e.  _V  ->  U.
ran  { A }  =  (/) )
189, 17eqtr4d 2318 . 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 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   {csn 3640   U.cuni 3827   ran crn 4690   ` cfv 5255   2ndc2nd 6121
This theorem is referenced by:  2nd0  6127  op2nd  6129  2nd2val  6146  elxp6  6151  2ndnpr  23246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-2nd 6123
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