Theorem fnop 5301

Description: The first argument of an ordered pair in a function belongs to the function's domain. (Contributed by NM, 8-Aug-1994.)

Assertion

Ref	Expression
fnop	|- ( ( F Fn A /\ <. B , C >. e. F ) -> B e. A )

Proof of Theorem fnop

Step	Hyp	Ref	Expression
1		df-br 3990	. 2 |- ( B F C <-> <. B , C >. e. F )
2		fnbr 5300	. 2 |- ( ( F Fn A /\ B F C ) -> B e. A )
3	1, 2	sylan2br 286	1 |- ( ( F Fn A /\ <. B , C >. e. F ) -> B e. A )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   <.cop 3586   class class class wbr 3989   Fn wfn 5193

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-dm 4621  df-fun 5200  df-fn 5201

This theorem is referenced by:  2elresin  5309  tfrlem9  6298  tfrexlem  6313