MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2ndval Structured version   Visualization version   Unicode version

Theorem 2ndval 6801
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 3980 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
21rneqd 5065 . . . 4  |-  ( x  =  A  ->  ran  { x }  =  ran  { A } )
32unieqd 4211 . . 3  |-  ( x  =  A  ->  U. ran  { x }  =  U. ran  { A } )
4 df-2nd 6799 . . 3  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
5 snex 4644 . . . . 5  |-  { A }  e.  _V
65rnex 6732 . . . 4  |-  ran  { A }  e.  _V
76uniex 6592 . . 3  |-  U. ran  { A }  e.  _V
83, 4, 7fvmpt 5953 . 2  |-  ( A  e.  _V  ->  ( 2nd `  A )  = 
U. ran  { A } )
9 fvprc 5864 . . 3  |-  ( -.  A  e.  _V  ->  ( 2nd `  A )  =  (/) )
10 snprc 4038 . . . . . . . 8  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1110biimpi 198 . . . . . . 7  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1211rneqd 5065 . . . . . 6  |-  ( -.  A  e.  _V  ->  ran 
{ A }  =  ran  (/) )
13 rn0 5089 . . . . . 6  |-  ran  (/)  =  (/)
1412, 13syl6eq 2503 . . . . 5  |-  ( -.  A  e.  _V  ->  ran 
{ A }  =  (/) )
1514unieqd 4211 . . . 4  |-  ( -.  A  e.  _V  ->  U.
ran  { A }  =  U. (/) )
16 uni0 4228 . . . 4  |-  U. (/)  =  (/)
1715, 16syl6eq 2503 . . 3  |-  ( -.  A  e.  _V  ->  U.
ran  { A }  =  (/) )
189, 17eqtr4d 2490 . 2  |-  ( -.  A  e.  _V  ->  ( 2nd `  A )  =  U. ran  { A } )
198, 18pm2.61i 168 1  |-  ( 2nd `  A )  =  U. ran  { A }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1446    e. wcel 1889   _Vcvv 3047   (/)c0 3733   {csn 3970   U.cuni 4201   ran crn 4838   ` cfv 5585   2ndc2nd 6797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5549  df-fun 5587  df-fv 5593  df-2nd 6799
This theorem is referenced by:  2ndnpr  6803  2nd0  6805  op2nd  6807  2nd2val  6825  elxp6  6830
  Copyright terms: Public domain W3C validator