Theorem fnbr 5363

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 5357	. . 3 |- ( F Fn A -> Rel F )
2		releldm 4902	. . 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 5358	. . . 4 |- ( F Fn A -> dom F = A )
5	4	eleq2d 2266	. . 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 2167   class class class wbr 4034   dom cdm 4664   Rel wrel 4669   Fn wfn 5254

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-xp 4670  df-rel 4671  df-dm 4674  df-fun 5261  df-fn 5262

This theorem is referenced by:  fnop  5364  dffn5im  5609  dffo4  5713  dffo5  5714  tfrlem5  6381