Theorem fnbr 3590

Description: The first argument of binary relation on a function belongs to the function's domain.

Assertion

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

Proof of Theorem fnbr

Step	Hyp	Ref	Expression
1		brrelex 3207	. . . 4 |- ((Rel F /\ BFC) -> B e. V)
2		fnrel 3586	. . . 4 |- (F Fn A -> Rel F)
3	1, 2	sylan 448	. . 3 |- ((F Fn A /\ BFC) -> B e. V)
4		breldmg 3316	. . 3 |- ((B e. V /\ BFC) -> B e. dom F)
5	3, 4	sylancom 475	. 2 |- ((F Fn A /\ BFC) -> B e. dom F)
6		fndm 3587	. . . 4 |- (F Fn A -> dom F = A)
7	6	eleq2d 1541	. . 3 |- (F Fn A -> (B e. dom F <-> B e. A))
8	7	biimpa 416	. 2 |- ((F Fn A /\ B e. dom F) -> B e. A)
9	5, 8	syldan 467	1 |- ((F Fn A /\ BFC) -> B e. A)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958  Vcvv 1811   class class class wbr 2619  dom cdm 3170  Rel wrel 3175   Fn wfn 3177

This theorem is referenced by:  fnop 3591  dffo4 3820  dffo5 3821

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-dm 3188  df-fun 3192  df-fn 3193

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