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Theorem opelrn 5806
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 5058 . 2 (𝐴𝐶𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶)
2 brelrn.1 . . 3 𝐴 ∈ V
3 brelrn.2 . . 3 𝐵 ∈ V
42, 3brelrn 5805 . 2 (𝐴𝐶𝐵𝐵 ∈ ran 𝐶)
51, 4sylbir 236 1 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐵 ∈ ran 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  Vcvv 3492  cop 4563   class class class wbr 5057  ran crn 5549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-cnv 5556  df-dm 5558  df-rn 5559
This theorem is referenced by:  dfres3  5851  zfrep6  7645  2ndrn  7729  disjen  8662  r0weon  9426  gsum2dlem1  19019  gsum2dlem2  19020  iss2  35482  rfovcnvf1od  40228
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