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Theorem opelrn 5953
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 5143 . 2 (𝐴𝐶𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶)
2 brelrn.1 . . 3 𝐴 ∈ V
3 brelrn.2 . . 3 𝐵 ∈ V
42, 3brelrn 5952 . 2 (𝐴𝐶𝐵𝐵 ∈ ran 𝐶)
51, 4sylbir 235 1 (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐵 ∈ ran 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3479  cop 4631   class class class wbr 5142  ran crn 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-cnv 5692  df-dm 5694  df-rn 5695
This theorem is referenced by:  dfres3  6001  relssdmrn  6287  zfrep6  7980  2ndrn  8067  disjen  9175  r0weon  10053  gsum2dlem1  19989  gsum2dlem2  19990  iss2  38346  rfovcnvf1od  44022
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