Theorem nfvres 6961

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 6041	. . . 4 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
2		inss1 4258	. . . 4 ⊢ (𝐵 ∩ dom 𝐹) ⊆ 𝐵
3	1, 2	eqsstri 4043	. . 3 ⊢ dom (𝐹 ↾ 𝐵) ⊆ 𝐵
4	3	sseli 4004	. 2 ⊢ (𝐴 ∈ dom (𝐹 ↾ 𝐵) → 𝐴 ∈ 𝐵)
5		ndmfv 6955	. 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 1537   ∈ wcel 2108   ∩ cin 3975  ∅c0 4352  dom cdm 5700   ↾ cres 5702  ‘cfv 6573

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-10 2141  ax-11 2158  ax-12 2178  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-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  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-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-dm 5710  df-res 5712  df-iota 6525  df-fv 6581

This theorem is referenced by:  fveqres  6967  fvresval  7394  funpartfv  35909  setrec2lem1  48785