Theorem fnresdisj 6658

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 5992	. . 3 ⊢ Rel (𝐹 ↾ 𝐵)
2		reldm0 5907	. . 3 ⊢ (Rel (𝐹 ↾ 𝐵) → ((𝐹 ↾ 𝐵) = ∅ ↔ dom (𝐹 ↾ 𝐵) = ∅))
3	1, 2	ax-mp 5	. 2 ⊢ ((𝐹 ↾ 𝐵) = ∅ ↔ dom (𝐹 ↾ 𝐵) = ∅)
4		dmres 5999	. . . . 5 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
5		incom 4184	. . . . 5 ⊢ (𝐵 ∩ dom 𝐹) = (dom 𝐹 ∩ 𝐵)
6	4, 5	eqtri 2758	. . . 4 ⊢ dom (𝐹 ↾ 𝐵) = (dom 𝐹 ∩ 𝐵)
7		fndm 6641	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
8	7	ineq1d 4194	. . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∩ 𝐵) = (𝐴 ∩ 𝐵))
9	6, 8	eqtrid 2782	. . 3 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ↾ 𝐵) = (𝐴 ∩ 𝐵))
10	9	eqeq1d 2737	. 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 3925  ∅c0 4308  dom cdm 5654   ↾ cres 5656  Rel wrel 5659   Fn wfn 6526

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-dm 5664  df-res 5666  df-fn 6534

This theorem is referenced by:  funressn  7149  fvsnun2  7175  dif1enlem  9170  dif1enlemOLD  9171  axdc3lem4  10467  fseq1p1m1  13615  hashgval  14351  hashinf  14353  pwssplit1  21017  mplmonmul  21994  wwlksm1edg  29863  eulerpartlemt  34403  poimirlem3  37647  pwssplit4  43113  isubgr0uhgr  47886