Theorem opelrn 5892

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 5080	. 2 ⊢ (𝐴𝐶𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶)
2		brelrn.1	. . 3 ⊢ 𝐴 ∈ V
3		brelrn.2	. . 3 ⊢ 𝐵 ∈ V
4	2, 3	brelrn 5891	. 2 ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶)
5	1, 4	sylbir 236	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐵 ∈ ran 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2119  Vcvv 3432  ⟨cop 4568   class class class wbr 5079  ran crn 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-cnv 5633  df-dm 5635  df-rn 5636

This theorem is referenced by:  dfres3  5943  relssdmrn  6227  zfrep6OLD  7904  2ndrn  7990  disjen  9069  r0weon  9932  gsum2dlem1  19943  gsum2dlem2  19944  iss2  38712  rfovcnvf1od  44449