Theorem opelrn 5933

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 5107	. 2 ⊢ (𝐴𝐶𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶)
2		brelrn.1	. . 3 ⊢ 𝐴 ∈ V
3		brelrn.2	. . 3 ⊢ 𝐵 ∈ V
4	2, 3	brelrn 5932	. 2 ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶)
5	1, 4	sylbir 238	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐵 ∈ ran 𝐶)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Vcvv 3454  ⟨cop 4838   class class class wbr 5106  ran crn 5657

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-cnv 5664  df-dm 5666  df-rn 5667

This theorem is used by:  dfres3  5987  relssdmrn  6290  zfrep6OLD  7994  2ndrn  8079  disjen  9184  r0weon  10121  gsum2dlem1  20217  gsum2dlem2  20218  iss2  39479  rfovcnvf1od  45207