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Theorem opelrn 5278
 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 4584 . 2 (𝐴𝐶𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶)
2 brelrn.1 . . 3 𝐴 ∈ V
3 brelrn.2 . . 3 𝐵 ∈ V
42, 3brelrn 5277 . 2 (𝐴𝐶𝐵𝐵 ∈ ran 𝐶)
51, 4sylbir 224 1 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐵 ∈ ran 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   class class class wbr 4583  ran crn 5039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-cnv 5046  df-dm 5048  df-rn 5049 This theorem is referenced by:  zfrep6  7027  2ndrn  7107  disjen  8002  r0weon  8718  gsum2dlem1  18192  gsum2dlem2  18193  dfres3  30902  rfovcnvf1od  37318
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