Theorem fvresval 7368

Description: The value of a restricted function at a class is either the empty set or the value of the unrestricted function at that class. (Contributed by Scott Fenton, 4-Sep-2011.)

Assertion

Ref	Expression
fvresval	⊢ (((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴) ∨ ((𝐹 ↾ 𝐵)‘𝐴) = ∅)

Proof of Theorem fvresval

Step	Hyp	Ref	Expression
1		exmid 908	. 2 ⊢ (𝐴 ∈ 𝐵 ∨ ¬ 𝐴 ∈ 𝐵)
2		fvres 6904	. . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
3		nfvres 6923	. . 3 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)
4	2, 3	orim12i 922	. 2 ⊢ ((𝐴 ∈ 𝐵 ∨ ¬ 𝐴 ∈ 𝐵) → (((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴) ∨ ((𝐹 ↾ 𝐵)‘𝐴) = ∅))
5	1, 4	ax-mp 5	1 ⊢ (((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴) ∨ ((𝐹 ↾ 𝐵)‘𝐴) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∅c0 4278   ↾ cres 5652  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-dm 5660  df-res 5662  df-iota 6494  df-fv 6546

This theorem is used by:  ltsres  28020