Theorem nfvres 6224

Description: The value of a non-member of a restriction is the empty set. (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 5419	. . . . 5 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
2		inss1 3833	. . . . 5 ⊢ (𝐵 ∩ dom 𝐹) ⊆ 𝐵
3	1, 2	eqsstri 3635	. . . 4 ⊢ dom (𝐹 ↾ 𝐵) ⊆ 𝐵
4	3	sseli 3599	. . 3 ⊢ (𝐴 ∈ dom (𝐹 ↾ 𝐵) → 𝐴 ∈ 𝐵)
5	4	con3i 150	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ dom (𝐹 ↾ 𝐵))
6		ndmfv 6218	. 2 ⊢ (¬ 𝐴 ∈ dom (𝐹 ↾ 𝐵) → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)
7	5, 6	syl 17	1 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1483   ∈ wcel 1990   ∩ cin 3573  ∅c0 3915  dom cdm 5114   ↾ cres 5116  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-dm 5124  df-res 5126  df-iota 5851  df-fv 5896

This theorem is referenced by:  fveqres  6230  fvresval  31665  trpredlem1  31727  funpartfv  32052