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

Theorem opelf 5752
 Description: The members of an ordered pair element of a mapping belong to the mapping's domain and codomain. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opelf

Proof of Theorem opelf
StepHypRef Expression
1 fssxp 5748 . . . 4
21sseld 3502 . . 3
3 opelxp 5034 . . 3
42, 3syl6ib 226 . 2
54imp 429 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  e.wcel 1818  <.cop 4035  X.cxp 5002  -->wf 5589 This theorem is referenced by:  feu  5766  fcnvres  5767  fsn  6069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-dm 5014  df-rn 5015  df-fun 5595  df-fn 5596  df-f 5597
 Copyright terms: Public domain W3C validator