Theorem opelrn 5943

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 5150	. 2 ⊢ (𝐴𝐶𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶)
2		brelrn.1	. . 3 ⊢ 𝐴 ∈ V
3		brelrn.2	. . 3 ⊢ 𝐵 ∈ V
4	2, 3	brelrn 5942	. 2 ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶)
5	1, 4	sylbir 234	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐵 ∈ ran 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2107  Vcvv 3475  ⟨cop 4635   class class class wbr 5149  ran crn 5678

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 5300  ax-nul 5307  ax-pr 5428

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 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-cnv 5685  df-dm 5687  df-rn 5688

This theorem is referenced by:  dfres3  5987  relssdmrn  6268  zfrep6  7941  2ndrn  8027  disjen  9134  r0weon  10007  gsum2dlem1  19838  gsum2dlem2  19839  iss2  37213  rfovcnvf1od  42755