Theorem opelrng 4731

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 3896	. 2 |- ( A C B <-> <. A , B >. e. C )
2		brelrng 4730	. 2 |- ( ( A e. F /\ B e. G /\ A C B ) -> B e. ran C )
3	1, 2	syl3an3br 1240	1 |- ( ( A e. F /\ B e. G /\ <. A , B >. e. C ) -> B e. ran C )

Syntax hints:   -> wi 4   /\ w3a 945   e. wcel 1463   <.cop 3496   class class class wbr 3895   ran crn 4500

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-cnv 4507  df-dm 4509  df-rn 4510

This theorem is referenced by:  2ndrn  6035