Theorem fnbr 5300

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 5296	. . 3 |- ( F Fn A -> Rel F )
2		releldm 4846	. . 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 5297	. . . 4 |- ( F Fn A -> dom F = A )
5	4	eleq2d 2240	. . 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 2141   class class class wbr 3989   dom cdm 4611   Rel wrel 4616   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:  fnop  5301  dffn5im  5542  dffo4  5644  dffo5  5645  tfrlem5  6293