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Theorem 2ndval 6352
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 3825 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
21rneqd 5097 . . . 4  |-  ( x  =  A  ->  ran  { x }  =  ran  { A } )
32unieqd 4026 . . 3  |-  ( x  =  A  ->  U. ran  { x }  =  U. ran  { A } )
4 df-2nd 6350 . . 3  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
5 snex 4405 . . . . 5  |-  { A }  e.  _V
65rnex 5133 . . . 4  |-  ran  { A }  e.  _V
76uniex 4705 . . 3  |-  U. ran  { A }  e.  _V
83, 4, 7fvmpt 5806 . 2  |-  ( A  e.  _V  ->  ( 2nd `  A )  = 
U. ran  { A } )
9 fvprc 5722 . . 3  |-  ( -.  A  e.  _V  ->  ( 2nd `  A )  =  (/) )
10 snprc 3871 . . . . . . . 8  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1110biimpi 187 . . . . . . 7  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1211rneqd 5097 . . . . . 6  |-  ( -.  A  e.  _V  ->  ran 
{ A }  =  ran  (/) )
13 rn0 5127 . . . . . 6  |-  ran  (/)  =  (/)
1412, 13syl6eq 2484 . . . . 5  |-  ( -.  A  e.  _V  ->  ran 
{ A }  =  (/) )
1514unieqd 4026 . . . 4  |-  ( -.  A  e.  _V  ->  U.
ran  { A }  =  U. (/) )
16 uni0 4042 . . . 4  |-  U. (/)  =  (/)
1715, 16syl6eq 2484 . . 3  |-  ( -.  A  e.  _V  ->  U.
ran  { A }  =  (/) )
189, 17eqtr4d 2471 . 2  |-  ( -.  A  e.  _V  ->  ( 2nd `  A )  =  U. ran  { A } )
198, 18pm2.61i 158 1  |-  ( 2nd `  A )  =  U. ran  { A }
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628   {csn 3814   U.cuni 4015   ran crn 4879   ` cfv 5454   2ndc2nd 6348
This theorem is referenced by:  2nd0  6354  op2nd  6356  2nd2val  6373  elxp6  6378  2ndnpr  24094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-2nd 6350
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