Theorem fnresdisj 6655

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 ∩ 𝐵) = ∅ ↔ (𝐹 ↾ 𝐵) = ∅))

Proof of Theorem fnresdisj

Step	Hyp	Ref	Expression
1		relres 6004	. . 3 ⊢ Rel (𝐹 ↾ 𝐵)
2		reldm0 5918	. . 3 ⊢ (Rel (𝐹 ↾ 𝐵) → ((𝐹 ↾ 𝐵) = ∅ ↔ dom (𝐹 ↾ 𝐵) = ∅))
3	1, 2	ax-mp 5	. 2 ⊢ ((𝐹 ↾ 𝐵) = ∅ ↔ dom (𝐹 ↾ 𝐵) = ∅)
4		dmres 6011	. . . . 5 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
5		incom 4162	. . . . 5 ⊢ (𝐵 ∩ dom 𝐹) = (dom 𝐹 ∩ 𝐵)
6	4, 5	eqtri 2786	. . . 4 ⊢ dom (𝐹 ↾ 𝐵) = (dom 𝐹 ∩ 𝐵)
7		fndm 6638	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
8	7	ineq1d 4172	. . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∩ 𝐵) = (𝐴 ∩ 𝐵))
9	6, 8	eqtrid 2810	. . 3 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ↾ 𝐵) = (𝐴 ∩ 𝐵))
10	9	eqeq1d 2765	. 2 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ↾ 𝐵) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅))
11	3, 10	bitr2id 287	1 ⊢ (𝐹 Fn 𝐴 → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐹 ↾ 𝐵) = ∅))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∩ cin 3904  ∅c0 4286  dom cdm 5661   ↾ cres 5663  Rel wrel 5666   Fn wfn 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-dm 5671  df-res 5673  df-fn 6539

This theorem is referenced by:  funressn  7156  fvsnun2  7181  dif1enlem  9140  axdc3lem4  10432  fseq1p1m1  13622  hashgval  14365  hashinf  14367  pwssplit1  21180  mplmonmul  22187  wwlksm1edg  30230  psrmonmul  33940  eulerpartlemt  34761  poimirlem3  38274  pwssplit4  43816  isubgr0uhgr  48638