Theorem opelrn 5880

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 5090	. 2 ⊢ (𝐴𝐶𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶)
2		brelrn.1	. . 3 ⊢ 𝐴 ∈ V
3		brelrn.2	. . 3 ⊢ 𝐵 ∈ V
4	2, 3	brelrn 5879	. 2 ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶)
5	1, 4	sylbir 235	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐵 ∈ ran 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2110  Vcvv 3434  ⟨cop 4580   class class class wbr 5089  ran crn 5615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-br 5090  df-opab 5152  df-cnv 5622  df-dm 5624  df-rn 5625

This theorem is referenced by:  dfres3  5930  relssdmrn  6212  zfrep6  7882  2ndrn  7968  disjen  9042  r0weon  9895  gsum2dlem1  19875  gsum2dlem2  19876  iss2  38351  rfovcnvf1od  44016