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Theorem opelrng 4741
Description: Membership of second member of an ordered pair in a range. (Contributed by Jim Kingdon, 26-Jan-2019.)
Assertion
Ref Expression
opelrng ((𝐴𝐹𝐵𝐺 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) → 𝐵 ∈ ran 𝐶)

Proof of Theorem opelrng
StepHypRef Expression
1 df-br 3900 . 2 (𝐴𝐶𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶)
2 brelrng 4740 . 2 ((𝐴𝐹𝐵𝐺𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)
31, 2syl3an3br 1242 1 ((𝐴𝐹𝐵𝐺 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) → 𝐵 ∈ ran 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 947  wcel 1465  cop 3500   class class class wbr 3899  ran crn 4510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-cnv 4517  df-dm 4519  df-rn 4520
This theorem is referenced by:  2ndrn  6049
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