Theorem fnresdisj 5308

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 4919	. . 3 |- Rel ( F |` B )
2		reldm0 4829	. . 3 |- ( Rel ( F |` B ) -> ( ( F |` B ) = (/) <-> dom ( F |` B ) = (/) ) )
3	1, 2	ax-mp 5	. 2 |- ( ( F |` B ) = (/) <-> dom ( F |` B ) = (/) )
4		dmres 4912	. . . . 5 |- dom ( F |` B ) = ( B i^i dom F )
5		incom 3319	. . . . 5 |- ( B i^i dom F ) = ( dom F i^i B )
6	4, 5	eqtri 2191	. . . 4 |- dom ( F |` B ) = ( dom F i^i B )
7		fndm 5297	. . . . 5 |- ( F Fn A -> dom F = A )
8	7	ineq1d 3327	. . . 4 |- ( F Fn A -> ( dom F i^i B ) = ( A i^i B ) )
9	6, 8	eqtrid 2215	. . 3 |- ( F Fn A -> dom ( F |` B ) = ( A i^i B ) )
10	9	eqeq1d 2179	. 2 |- ( F Fn A -> ( dom ( F |` B ) = (/) <-> ( A i^i B ) = (/) ) )
11	3, 10	bitr2id 192	1 |- ( F Fn A -> ( ( A i^i B ) = (/) <-> ( F |` B ) = (/) ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   i^i cin 3120   (/)c0 3414   dom cdm 4611   |` cres 4613   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-in1 609  ax-in2 610  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-fal 1354  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-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  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-res 4623  df-fn 5201

This theorem is referenced by:  fvsnun2  5694  fseq1p1m1  10050