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

Proof of Theorem 2ndval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sneq 3980 . . . . 5
21rneqd 5065 . . . 4
32unieqd 4211 . . 3
4 df-2nd 6799 . . 3
5 snex 4644 . . . . 5
65rnex 6732 . . . 4
76uniex 6592 . . 3
83, 4, 7fvmpt 5953 . 2
9 fvprc 5864 . . 3
10 snprc 4038 . . . . . . . 8
1110biimpi 198 . . . . . . 7
1211rneqd 5065 . . . . . 6
13 rn0 5089 . . . . . 6
1412, 13syl6eq 2503 . . . . 5
1514unieqd 4211 . . . 4
16 uni0 4228 . . . 4
1715, 16syl6eq 2503 . . 3
189, 17eqtr4d 2490 . 2
198, 18pm2.61i 168 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wceq 1446   wcel 1889  cvv 3047  c0 3733  csn 3970  cuni 4201   crn 4838  cfv 5585  c2nd 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