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Theorem fvresval 33018
 Description: The value of a function at a restriction is either null or the same as the function itself. (Contributed by Scott Fenton, 4-Sep-2011.)
Assertion
Ref Expression
fvresval (((𝐹𝐵)‘𝐴) = (𝐹𝐴) ∨ ((𝐹𝐵)‘𝐴) = ∅)

Proof of Theorem fvresval
StepHypRef Expression
1 exmid 891 . 2 (𝐴𝐵 ∨ ¬ 𝐴𝐵)
2 fvres 6665 . . 3 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
3 nfvres 6682 . . 3 𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)
42, 3orim12i 905 . 2 ((𝐴𝐵 ∨ ¬ 𝐴𝐵) → (((𝐹𝐵)‘𝐴) = (𝐹𝐴) ∨ ((𝐹𝐵)‘𝐴) = ∅))
51, 4ax-mp 5 1 (((𝐹𝐵)‘𝐴) = (𝐹𝐴) ∨ ((𝐹𝐵)‘𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 843   = wceq 1537   ∈ wcel 2114  ∅c0 4269   ↾ cres 5533  ‘cfv 6331 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-xp 5537  df-dm 5541  df-res 5543  df-iota 6290  df-fv 6339 This theorem is referenced by:  sltres  33177
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