Theorem fnresdisj 2732

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

Assertion

Ref	Expression
fnresdisj	⊢ (F Fn A → ((A ∩ B) = ∅ ↔ (F ↾ B) = ∅))

Proof of Theorem fnresdisj

Step	Hyp	Ref	Expression
1		fndm 2723	. . . . 5 ⊢ (F Fn A → dom F = A)
2	1	ineq1d 1644	. . . 4 ⊢ (F Fn A → (dom F ∩ B) = (A ∩ B))
3		dmres 2584	. . . . 5 ⊢ dom (F ↾ B) = (B ∩ dom F)
4		incom 1636	. . . . 5 ⊢ (B ∩ dom F) = (dom F ∩ B)
5	3, 4	eqtr 1119	. . . 4 ⊢ dom (F ↾ B) = (dom F ∩ B)
6	2, 5	syl5eq 1136	. . 3 ⊢ (F Fn A → dom (F ↾ B) = (A ∩ B))
7	6	cleq1d 1109	. 2 ⊢ (F Fn A → (dom (F ↾ B) = ∅ ↔ (A ∩ B) = ∅))
8		relres 2591	. . 3 ⊢ Rel (F ↾ B)
9		reldm0 2550	. . 3 ⊢ (Rel (F ↾ B) → ((F ↾ B) = ∅ ↔ dom (F ↾ B) = ∅))
10	8, 9	ax-mp 6	. 2 ⊢ ((F ↾ B) = ∅ ↔ dom (F ↾ B) = ∅)
11	7, 10	syl5rbb 411	1 ⊢ (F Fn A → ((A ∩ B) = ∅ ↔ (F ↾ B) = ∅))

Syntax hints:   → wi 2   ↔ wb 127   = wceq 1091   ∩ cin 1486  ∅c0 1707  dom cdm 2410   ↾ cres 2412  Rel wrel 2415   Fn wfn 2417

This theorem is referenced by:  mapunen 3397  facnnt 4870  fac0 4871  ruclem6 4890

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-dm 2428  df-res 2430  df-fn 2433

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