Theorem fnresdisj 5009

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 4639	. . 3 |- Rel ( F |` B )
2		reldm0 4553	. . 3 |- ( Rel ( F |` B ) -> ( ( F |` B ) = (/) <-> dom ( F |` B ) = (/) ) )
3	1, 2	ax-mp 7	. 2 |- ( ( F |` B ) = (/) <-> dom ( F |` B ) = (/) )
4		dmres 4632	. . . . 5 |- dom ( F |` B ) = ( B i^i dom F )
5		incom 3129	. . . . 5 |- ( B i^i dom F ) = ( dom F i^i B )
6	4, 5	eqtri 2060	. . . 4 |- dom ( F |` B ) = ( dom F i^i B )
7		fndm 4998	. . . . 5 |- ( F Fn A -> dom F = A )
8	7	ineq1d 3137	. . . 4 |- ( F Fn A -> ( dom F i^i B ) = ( A i^i B ) )
9	6, 8	syl5eq 2084	. . 3 |- ( F Fn A -> dom ( F |` B ) = ( A i^i B ) )
10	9	eqeq1d 2048	. 2 |- ( F Fn A -> ( dom ( F |` B ) = (/) <-> ( A i^i B ) = (/) ) )
11	3, 10	syl5rbb 182	1 |- ( F Fn A -> ( ( A i^i B ) = (/) <-> ( F |` B ) = (/) ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   i^i cin 2916   (/)c0 3224   dom cdm 4345   |` cres 4347   Rel wrel 4350   Fn wfn 4897

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-dm 4355  df-res 4357  df-fn 4905

This theorem is referenced by:  fvsnun2  5361  fseq1p1m1  8956