Theorem fnresdisj 6596

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 5949	. . 3 ⊢ Rel (𝐹 ↾ 𝐵)
2		reldm0 5863	. . 3 ⊢ (Rel (𝐹 ↾ 𝐵) → ((𝐹 ↾ 𝐵) = ∅ ↔ dom (𝐹 ↾ 𝐵) = ∅))
3	1, 2	ax-mp 5	. 2 ⊢ ((𝐹 ↾ 𝐵) = ∅ ↔ dom (𝐹 ↾ 𝐵) = ∅)
4		dmres 5956	. . . . 5 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
5		incom 4154	. . . . 5 ⊢ (𝐵 ∩ dom 𝐹) = (dom 𝐹 ∩ 𝐵)
6	4, 5	eqtri 2754	. . . 4 ⊢ dom (𝐹 ↾ 𝐵) = (dom 𝐹 ∩ 𝐵)
7		fndm 6579	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
8	7	ineq1d 4164	. . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∩ 𝐵) = (𝐴 ∩ 𝐵))
9	6, 8	eqtrid 2778	. . 3 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ↾ 𝐵) = (𝐴 ∩ 𝐵))
10	9	eqeq1d 2733	. 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 1541   ∩ cin 3896  ∅c0 4278  dom cdm 5611   ↾ cres 5613  Rel wrel 5616   Fn wfn 6471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149  df-xp 5617  df-rel 5618  df-dm 5621  df-res 5623  df-fn 6479

This theorem is referenced by:  funressn  7087  fvsnun2  7112  dif1enlem  9064  axdc3lem4  10339  fseq1p1m1  13493  hashgval  14235  hashinf  14237  pwssplit1  20988  mplmonmul  21966  wwlksm1edg  29854  eulerpartlemt  34376  poimirlem3  37663  pwssplit4  43122  isubgr0uhgr  47904