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Theorem fvresval 7351
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 (((𝐹𝐵)‘𝐴) = (𝐹𝐴) ∨ ((𝐹𝐵)‘𝐴) = ∅)

Proof of Theorem fvresval
StepHypRef Expression
1 exmid 891 . 2 (𝐴𝐵 ∨ ¬ 𝐴𝐵)
2 fvres 6904 . . 3 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
3 nfvres 6926 . . 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 844   = wceq 1533  wcel 2098  c0 4317  cres 5671  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-dm 5679  df-res 5681  df-iota 6489  df-fv 6545
This theorem is referenced by:  sltres  27550
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