Theorem fnresdisj 3603

Description: A function restricted to a class disjoint with its domain is empty.

Assertion

Ref	Expression
fnresdisj	|- (F Fn A -> ((A i^i B) = (/) <-> (F |` B) = (/)))

Proof of Theorem fnresdisj

Step	Hyp	Ref	Expression
1		fndm 3593	. . . . 5 |- (F Fn A -> dom F = A)
2	1	ineq1d 2219	. . . 4 |- (F Fn A -> (dom F i^i B) = (A i^i B))
3		dmres 3386	. . . . 5 |- dom ( F |` B) = (B i^i dom F)
4		incom 2211	. . . . 5 |- (B i^i dom F) = (dom F i^i B)
5	3, 4	eqtr 1498	. . . 4 |- dom ( F |` B) = (dom F i^i B)
6	2, 5	syl5eq 1522	. . 3 |- (F Fn A -> dom ( F |` B) = (A i^i B))
7	6	eqeq1d 1486	. 2 |- (F Fn A -> (dom ( F |` B) = (/) <-> (A i^i B) = (/)))
8		relres 3393	. . 3 |- Rel (F |` B)
9		reldm0 3337	. . 3 |- (Rel (F |` B) -> ((F |` B) = (/) <-> dom ( F |` B) = (/)))
10	8, 9	ax-mp 7	. 2 |- ((F |` B) = (/) <-> dom ( F |` B) = (/))
11	7, 10	syl5rbb 535	1 |- (F Fn A -> ((A i^i B) = (/) <-> (F |` B) = (/)))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   i^i cin 2049  (/)c0 2283  dom cdm 3176   |` cres 3178  Rel wrel 3181   Fn wfn 3183

This theorem is referenced by:  fvsnun2 3802  mapunen 4508  acdc2lem2 7490  acdc5lem2 7493  ruclem6 7516

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-dm 3194  df-res 3196  df-fn 3199

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