Theorem fnresdisj 3597

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 3587	. . . . 5 |- (F Fn A -> dom F = A)
2	1	ineq1d 2216	. . . 4 |- (F Fn A -> (dom F i^i B) = (A i^i B))
3		dmres 3380	. . . . 5 |- dom ( F |` B) = (B i^i dom F)
4		incom 2208	. . . . 5 |- (B i^i dom F) = (dom F i^i B)
5	3, 4	eqtr 1495	. . . 4 |- dom ( F |` B) = (dom F i^i B)
6	2, 5	syl5eq 1519	. . 3 |- (F Fn A -> dom ( F |` B) = (A i^i B))
7	6	eqeq1d 1483	. 2 |- (F Fn A -> (dom ( F |` B) = (/) <-> (A i^i B) = (/)))
8		relres 3387	. . 3 |- Rel (F |` B)
9		reldm0 3331	. . 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 533	1 |- (F Fn A -> ((A i^i B) = (/) <-> (F |` B) = (/)))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   i^i cin 2046  (/)c0 2280  dom cdm 3170   |` cres 3172  Rel wrel 3175   Fn wfn 3177

This theorem is referenced by:  fvsnun2 3796  mapunen 4502  acdc2lem2 7489  acdc5lem2 7492  ruclem6 7515

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-dm 3188  df-res 3190  df-fn 3193

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