Theorem opelrng 4996

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 4116	. 2 |- ( A C B <-> <. A , B >. e. C )
2		brelrng 4995	. 2 |- ( ( A e. F /\ B e. G /\ A C B ) -> B e. ran C )
3	1, 2	syl3an3br 1315	1 |- ( ( A e. F /\ B e. G /\ <. A , B >. e. C ) -> B e. ran C )

Syntax hints:   -> wi 4   /\ w3a 1005   e. wcel 2205   <.cop 3698   class class class wbr 4115   ran crn 4757

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-br 4116  df-opab 4178  df-cnv 4764  df-dm 4766  df-rn 4767

This theorem is referenced by:  2ndrn  6392