Theorem fnresdisj 5368

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 4974	. . 3 |- Rel ( F |` B )
2		reldm0 4884	. . 3 |- ( Rel ( F |` B ) -> ( ( F |` B ) = (/) <-> dom ( F |` B ) = (/) ) )
3	1, 2	ax-mp 5	. 2 |- ( ( F |` B ) = (/) <-> dom ( F |` B ) = (/) )
4		dmres 4967	. . . . 5 |- dom ( F |` B ) = ( B i^i dom F )
5		incom 3355	. . . . 5 |- ( B i^i dom F ) = ( dom F i^i B )
6	4, 5	eqtri 2217	. . . 4 |- dom ( F |` B ) = ( dom F i^i B )
7		fndm 5357	. . . . 5 |- ( F Fn A -> dom F = A )
8	7	ineq1d 3363	. . . 4 |- ( F Fn A -> ( dom F i^i B ) = ( A i^i B ) )
9	6, 8	eqtrid 2241	. . 3 |- ( F Fn A -> dom ( F |` B ) = ( A i^i B ) )
10	9	eqeq1d 2205	. 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 1364   i^i cin 3156   (/)c0 3450   dom cdm 4663   |` cres 4665   Rel wrel 4668   Fn wfn 5253

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-dm 4673  df-res 4675  df-fn 5261

This theorem is referenced by:  fvsnun2  5760  fseq1p1m1  10169