Theorem fnbr 5425

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 5419	. . 3 |- ( F Fn A -> Rel F )
2		releldm 4959	. . 3 |- ( ( Rel F /\ B F C ) -> B e. dom F )
3	1, 2	sylan 283	. 2 |- ( ( F Fn A /\ B F C ) -> B e. dom F )
4		fndm 5420	. . . 4 |- ( F Fn A -> dom F = A )
5	4	eleq2d 2299	. . 3 |- ( F Fn A -> ( B e. dom F <-> B e. A ) )
6	5	biimpa 296	. 2 |- ( ( F Fn A /\ B e. dom F ) -> B e. A )
7	3, 6	syldan 282	1 |- ( ( F Fn A /\ B F C ) -> B e. A )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   class class class wbr 4083   dom cdm 4719   Rel wrel 4724   Fn wfn 5313

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-dm 4729  df-fun 5320  df-fn 5321

This theorem is referenced by:  fnop  5426  dffn5im  5679  dffo4  5783  dffo5  5784  tfrlem5  6460