Theorem fnbr 5290

Description: The first argument of binary relation on a function belongs to the function's domain. (Contributed by NM, 7-May-2004.)

Assertion

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

Proof of Theorem fnbr

Step	Hyp	Ref	Expression
1		fnrel 5286	. . 3 |- ( F Fn A -> Rel F )
2		releldm 4839	. . 3 |- ( ( Rel F /\ B F C ) -> B e. dom F )
3	1, 2	sylan 281	. 2 |- ( ( F Fn A /\ B F C ) -> B e. dom F )
4		fndm 5287	. . . 4 |- ( F Fn A -> dom F = A )
5	4	eleq2d 2236	. . 3 |- ( F Fn A -> ( B e. dom F <-> B e. A ) )
6	5	biimpa 294	. 2 |- ( ( F Fn A /\ B e. dom F ) -> B e. A )
7	3, 6	syldan 280	1 |- ( ( F Fn A /\ B F C ) -> B e. A )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   class class class wbr 3982   dom cdm 4604   Rel wrel 4609   Fn wfn 5183

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-dm 4614  df-fun 5190  df-fn 5191

This theorem is referenced by:  fnop  5291  dffn5im  5532  dffo4  5633  dffo5  5634  tfrlem5  6282