Theorem fnresdisj 5671

Description: A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)

Assertion

Ref	Expression
fnresdisj	|- ( F Fn A -> ( ( A i^i B ) = (/) <-> ( F |` B ) = (/) ) )

Proof of Theorem fnresdisj

Step	Hyp	Ref	Expression
1		relres 5120	. . 3 |- Rel ( F |` B )
2		reldm0 5040	. . 3 |- ( Rel ( F |` B ) -> ( ( F |` B ) = (/) <-> dom ( F |` B ) = (/) ) )
3	1, 2	ax-mp 5	. 2 |- ( ( F |` B ) = (/) <-> dom ( F |` B ) = (/) )
4		dmres 5113	. . . . 5 |- dom ( F |` B ) = ( B i^i dom F )
5		incom 3631	. . . . 5 |- ( B i^i dom F ) = ( dom F i^i B )
6	4, 5	eqtri 2431	. . . 4 |- dom ( F |` B ) = ( dom F i^i B )
7		fndm 5660	. . . . 5 |- ( F Fn A -> dom F = A )
8	7	ineq1d 3639	. . . 4 |- ( F Fn A -> ( dom F i^i B ) = ( A i^i B ) )
9	6, 8	syl5eq 2455	. . 3 |- ( F Fn A -> dom ( F |` B ) = ( A i^i B ) )
10	9	eqeq1d 2404	. 2 |- ( F Fn A -> ( dom ( F |` B ) = (/) <-> ( A i^i B ) = (/) ) )
11	3, 10	syl5rbb 258	1 |- ( F Fn A -> ( ( A i^i B ) = (/) <-> ( F |` B ) = (/) ) )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1405   i^i cin 3412   (/)c0 3737   dom cdm 4822   |` cres 4824   Rel wrel 4827   Fn wfn 5563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4828  df-rel 4829  df-dm 4832  df-res 4834  df-fn 5571

This theorem is referenced by:  funressn  6063  fvsnun2  6086  axdc3lem4  8864  fseq1p1m1  11805  hashgval  12453  hashinf  12455  pwssplit1  18023  mplmonmul  18444  wwlkm1edg  25139  eulerpartlemt  28802  pwssplit4  35377