Theorem opelrng 4874

Description: Membership of second member of an ordered pair in a range. (Contributed by Jim Kingdon, 26-Jan-2019.)

Assertion

Ref	Expression
opelrng	⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) → 𝐵 ∈ ran 𝐶)

Proof of Theorem opelrng

Step	Hyp	Ref	Expression
1		df-br 4019	. 2 ⊢ (𝐴𝐶𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶)
2		brelrng 4873	. 2 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)
3	1, 2	syl3an3br 1290	1 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) → 𝐵 ∈ ran 𝐶)

Syntax hints:   → wi 4   ∧ w3a 980   ∈ wcel 2160  ⟨cop 3610   class class class wbr 4018  ran crn 4642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-cnv 4649  df-dm 4651  df-rn 4652

This theorem is referenced by:  2ndrn  6202