Theorem fnbr 5379

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 5373	. . 3 |- ( F Fn A -> Rel F )
2		releldm 4914	. . 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 5374	. . . 4 |- ( F Fn A -> dom F = A )
5	4	eleq2d 2275	. . 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 2176   class class class wbr 4045   dom cdm 4676   Rel wrel 4681   Fn wfn 5267

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-xp 4682  df-rel 4683  df-dm 4686  df-fun 5274  df-fn 5275

This theorem is referenced by:  fnop  5380  dffn5im  5626  dffo4  5730  dffo5  5731  tfrlem5  6402