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Theorem tz6.12-2 6094
 Description: Function value when 𝐹 is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
tz6.12-2 (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = ∅)
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem tz6.12-2
StepHypRef Expression
1 df-fv 5812 . 2 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
2 iotanul 5783 . 2 (¬ ∃!𝑥 𝐴𝐹𝑥 → (℩𝑥𝐴𝐹𝑥) = ∅)
31, 2syl5eq 2656 1 (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475  ∃!weu 2458  ∅c0 3874   class class class wbr 4583  ℩cio 5766  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-uni 4373  df-iota 5768  df-fv 5812 This theorem is referenced by:  fvprc  6097  tz6.12i  6124  ndmfv  6128  nfunsn  6135  funpartfv  31222  setrec2lem1  42239
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