Theorem fnresdisj 5328

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 4937	. . 3 |- Rel ( F |` B )
2		reldm0 4847	. . 3 |- ( Rel ( F |` B ) -> ( ( F |` B ) = (/) <-> dom ( F |` B ) = (/) ) )
3	1, 2	ax-mp 5	. 2 |- ( ( F |` B ) = (/) <-> dom ( F |` B ) = (/) )
4		dmres 4930	. . . . 5 |- dom ( F |` B ) = ( B i^i dom F )
5		incom 3329	. . . . 5 |- ( B i^i dom F ) = ( dom F i^i B )
6	4, 5	eqtri 2198	. . . 4 |- dom ( F |` B ) = ( dom F i^i B )
7		fndm 5317	. . . . 5 |- ( F Fn A -> dom F = A )
8	7	ineq1d 3337	. . . 4 |- ( F Fn A -> ( dom F i^i B ) = ( A i^i B ) )
9	6, 8	eqtrid 2222	. . 3 |- ( F Fn A -> dom ( F |` B ) = ( A i^i B ) )
10	9	eqeq1d 2186	. 2 |- ( F Fn A -> ( dom ( F |` B ) = (/) <-> ( A i^i B ) = (/) ) )
11	3, 10	bitr2id 193	1 |- ( F Fn A -> ( ( A i^i B ) = (/) <-> ( F |` B ) = (/) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   i^i cin 3130   (/)c0 3424   dom cdm 4628   |` cres 4630   Rel wrel 4633   Fn wfn 5213

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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-dm 4638  df-res 4640  df-fn 5221

This theorem is referenced by:  fvsnun2  5716  fseq1p1m1  10096