Theorem fnresdisj 6700

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 6035	. . 3 ⊢ Rel (𝐹 ↾ 𝐵)
2		reldm0 5952	. . 3 ⊢ (Rel (𝐹 ↾ 𝐵) → ((𝐹 ↾ 𝐵) = ∅ ↔ dom (𝐹 ↾ 𝐵) = ∅))
3	1, 2	ax-mp 5	. 2 ⊢ ((𝐹 ↾ 𝐵) = ∅ ↔ dom (𝐹 ↾ 𝐵) = ∅)
4		dmres 6041	. . . . 5 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
5		incom 4230	. . . . 5 ⊢ (𝐵 ∩ dom 𝐹) = (dom 𝐹 ∩ 𝐵)
6	4, 5	eqtri 2768	. . . 4 ⊢ dom (𝐹 ↾ 𝐵) = (dom 𝐹 ∩ 𝐵)
7		fndm 6682	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
8	7	ineq1d 4240	. . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∩ 𝐵) = (𝐴 ∩ 𝐵))
9	6, 8	eqtrid 2792	. . 3 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ↾ 𝐵) = (𝐴 ∩ 𝐵))
10	9	eqeq1d 2742	. 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 1537   ∩ cin 3975  ∅c0 4352  dom cdm 5700   ↾ cres 5702  Rel wrel 5705   Fn wfn 6568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-dm 5710  df-res 5712  df-fn 6576

This theorem is referenced by:  funressn  7193  fvsnun2  7217  dif1enlem  9222  dif1enlemOLD  9223  axdc3lem4  10522  fseq1p1m1  13658  hashgval  14382  hashinf  14384  pwssplit1  21081  mplmonmul  22077  wwlksm1edg  29914  eulerpartlemt  34336  poimirlem3  37583  pwssplit4  43046  isubgr0uhgr  47743