Theorem nfvres 6906

Description: The value of a non-member of a restriction is the empty set. (An artifact of our function value definition.) (Contributed by NM, 13-Nov-1995.)

Assertion

Ref	Expression
nfvres	⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)

Proof of Theorem nfvres

Step	Hyp	Ref	Expression
1		dmres 5999	. . . 4 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
2		inss1 4189	. . . 4 ⊢ (𝐵 ∩ dom 𝐹) ⊆ 𝐵
3	1, 2	eqsstri 3983	. . 3 ⊢ dom (𝐹 ↾ 𝐵) ⊆ 𝐵
4	3	sseli 3933	. 2 ⊢ (𝐴 ∈ dom (𝐹 ↾ 𝐵) → 𝐴 ∈ 𝐵)
5		ndmfv 6900	. 2 ⊢ (¬ 𝐴 ∈ dom (𝐹 ↾ 𝐵) → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)
6	4, 5	nsyl5 159	1 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1561   ∈ wcel 2143   ∩ cin 3904  ∅c0 4286  dom cdm 5648   ↾ cres 5650  ‘cfv 6522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-xp 5654  df-dm 5658  df-res 5660  df-iota 6478  df-fv 6530

This theorem is referenced by:  fveqres  6912  fvresval  7343  funpartfv  36296  setrec2lem1  50315