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Theorem opelrn 5899
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 5107 . 2 (𝐴𝐶𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶)
2 brelrn.1 . . 3 𝐴 ∈ V
3 brelrn.2 . . 3 𝐵 ∈ V
42, 3brelrn 5898 . 2 (𝐴𝐶𝐵𝐵 ∈ ran 𝐶)
51, 4sylbir 234 1 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐵 ∈ ran 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3444  cop 4593   class class class wbr 5106  ran crn 5635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-cnv 5642  df-dm 5644  df-rn 5645
This theorem is referenced by:  dfres3  5943  relssdmrn  6221  zfrep6  7888  2ndrn  7974  disjen  9081  r0weon  9953  gsum2dlem1  19752  gsum2dlem2  19753  iss2  36851  rfovcnvf1od  42364
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