Theorem fnresdisj 5704

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 5151	. . 3 |- Rel ( F |` B )
2		reldm0 5071	. . 3 |- ( Rel ( F |` B ) -> ( ( F |` B ) = (/) <-> dom ( F |` B ) = (/) ) )
3	1, 2	ax-mp 5	. 2 |- ( ( F |` B ) = (/) <-> dom ( F |` B ) = (/) )
4		dmres 5144	. . . . 5 |- dom ( F |` B ) = ( B i^i dom F )
5		incom 3655	. . . . 5 |- ( B i^i dom F ) = ( dom F i^i B )
6	4, 5	eqtri 2451	. . . 4 |- dom ( F |` B ) = ( dom F i^i B )
7		fndm 5693	. . . . 5 |- ( F Fn A -> dom F = A )
8	7	ineq1d 3663	. . . 4 |- ( F Fn A -> ( dom F i^i B ) = ( A i^i B ) )
9	6, 8	syl5eq 2475	. . 3 |- ( F Fn A -> dom ( F |` B ) = ( A i^i B ) )
10	9	eqeq1d 2424	. 2 |- ( F Fn A -> ( dom ( F |` B ) = (/) <-> ( A i^i B ) = (/) ) )
11	3, 10	syl5rbb 261	1 |- ( F Fn A -> ( ( A i^i B ) = (/) <-> ( F |` B ) = (/) ) )

Syntax hints:   -> wi 4   <-> wb 187   = wceq 1437   i^i cin 3435   (/)c0 3761   dom cdm 4853   |` cres 4855   Rel wrel 4858   Fn wfn 5596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483  df-xp 4859  df-rel 4860  df-dm 4863  df-res 4865  df-fn 5604

This theorem is referenced by:  funressn  6092  fvsnun2  6115  axdc3lem4  8890  fseq1p1m1  11875  hashgval  12524  hashinf  12526  pwssplit1  18281  mplmonmul  18687  wwlkm1edg  25461  eulerpartlemt  29212  poimirlem3  31907  pwssplit4  35917