Theorem opelrng 4884

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

Assertion

Ref	Expression
opelrng	|- ( ( A e. F /\ B e. G /\ <. A , B >. e. C ) -> B e. ran C )

Proof of Theorem opelrng

Step	Hyp	Ref	Expression
1		df-br 4026	. 2 |- ( A C B <-> <. A , B >. e. C )
2		brelrng 4883	. 2 |- ( ( A e. F /\ B e. G /\ A C B ) -> B e. ran C )
3	1, 2	syl3an3br 1290	1 |- ( ( A e. F /\ B e. G /\ <. A , B >. e. C ) -> B e. ran C )

Syntax hints:   -> wi 4   /\ w3a 980   e. wcel 2160   <.cop 3617   class class class wbr 4025   ran crn 4652

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 4143  ax-pow 4199  ax-pr 4234

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 2758  df-un 3152  df-in 3154  df-ss 3161  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-br 4026  df-opab 4087  df-cnv 4659  df-dm 4661  df-rn 4662

This theorem is referenced by:  2ndrn  6216