MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opelrn Structured version   Visualization version   GIF version

Theorem opelrn 5494
Description: Membership of second member of an ordered pair in a range. (Contributed by NM, 23-Feb-1997.)
Hypotheses
Ref Expression
brelrn.1 𝐴 ∈ V
brelrn.2 𝐵 ∈ V
Assertion
Ref Expression
opelrn (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐵 ∈ ran 𝐶)

Proof of Theorem opelrn
StepHypRef Expression
1 df-br 4788 . 2 (𝐴𝐶𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶)
2 brelrn.1 . . 3 𝐴 ∈ V
3 brelrn.2 . . 3 𝐵 ∈ V
42, 3brelrn 5493 . 2 (𝐴𝐶𝐵𝐵 ∈ ran 𝐶)
51, 4sylbir 225 1 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐵 ∈ ran 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  Vcvv 3351  cop 4323   class class class wbr 4787  ran crn 5251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-cnv 5258  df-dm 5260  df-rn 5261
This theorem is referenced by:  dfres3  5538  zfrep6  7285  2ndrn  7369  disjen  8277  r0weon  9039  gsum2dlem1  18576  gsum2dlem2  18577  cnfinltrel  33577  iss2  34452  rfovcnvf1od  38822
  Copyright terms: Public domain W3C validator