Theorem opelrn 5841

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⟩ ∈ 𝐶 → 𝐵 ∈ ran 𝐶)

Proof of Theorem opelrn

Step	Hyp	Ref	Expression
1		df-br 5071	. 2 ⊢ (𝐴𝐶𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶)
2		brelrn.1	. . 3 ⊢ 𝐴 ∈ V
3		brelrn.2	. . 3 ⊢ 𝐵 ∈ V
4	2, 3	brelrn 5840	. 2 ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶)
5	1, 4	sylbir 234	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐵 ∈ ran 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2108  Vcvv 3422  ⟨cop 4564   class class class wbr 5070  ran crn 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-cnv 5588  df-dm 5590  df-rn 5591

This theorem is referenced by:  dfres3  5885  zfrep6  7771  2ndrn  7855  disjen  8870  r0weon  9699  gsum2dlem1  19486  gsum2dlem2  19487  iss2  36406  rfovcnvf1od  41501