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

Theorem 2ndrn 6184
 Description: The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.)
Assertion
Ref Expression
2ndrn

Proof of Theorem 2ndrn
StepHypRef Expression
1 1st2nd 6182 . . 3
2 simpr 447 . . 3
31, 2eqeltrrd 2371 . 2
4 fvex 5555 . . 3
5 fvex 5555 . . 3
64, 5opelrn 4926 . 2
73, 6syl 15 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wcel 1696  cop 3656   crn 4706   wrel 4710  cfv 5271  c1st 6136  c2nd 6137 This theorem is referenced by:  11st22nd  25148 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-1st 6138  df-2nd 6139
 Copyright terms: Public domain W3C validator