Theorem fnresdisj 6606

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 5960	. . 3 ⊢ Rel (𝐹 ↾ 𝐵)
2		reldm0 5874	. . 3 ⊢ (Rel (𝐹 ↾ 𝐵) → ((𝐹 ↾ 𝐵) = ∅ ↔ dom (𝐹 ↾ 𝐵) = ∅))
3	1, 2	ax-mp 5	. 2 ⊢ ((𝐹 ↾ 𝐵) = ∅ ↔ dom (𝐹 ↾ 𝐵) = ∅)
4		dmres 5967	. . . . 5 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
5		incom 4162	. . . . 5 ⊢ (𝐵 ∩ dom 𝐹) = (dom 𝐹 ∩ 𝐵)
6	4, 5	eqtri 2752	. . . 4 ⊢ dom (𝐹 ↾ 𝐵) = (dom 𝐹 ∩ 𝐵)
7		fndm 6589	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
8	7	ineq1d 4172	. . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∩ 𝐵) = (𝐴 ∩ 𝐵))
9	6, 8	eqtrid 2776	. . 3 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ↾ 𝐵) = (𝐴 ∩ 𝐵))
10	9	eqeq1d 2731	. 2 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ↾ 𝐵) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅))
11	3, 10	bitr2id 284	1 ⊢ (𝐹 Fn 𝐴 → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐹 ↾ 𝐵) = ∅))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∩ cin 3904  ∅c0 4286  dom cdm 5623   ↾ cres 5625  Rel wrel 5628   Fn wfn 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-rel 5630  df-dm 5633  df-res 5635  df-fn 6489

This theorem is referenced by:  funressn  7097  fvsnun2  7123  dif1enlem  9080  dif1enlemOLD  9081  axdc3lem4  10366  fseq1p1m1  13520  hashgval  14259  hashinf  14261  pwssplit1  20982  mplmonmul  21960  wwlksm1edg  29845  eulerpartlemt  34358  poimirlem3  37622  pwssplit4  43082  isubgr0uhgr  47877