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Theorem tz6.12-2 6323
 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 6039 . 2 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
2 iotanul 6009 . 2 (¬ ∃!𝑥 𝐴𝐹𝑥 → (℩𝑥𝐴𝐹𝑥) = ∅)
31, 2syl5eq 2816 1 (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1630  ∃!weu 2617  ∅c0 4061   class class class wbr 4784  ℩cio 5992  ‘cfv 6031 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-v 3351  df-dif 3724  df-in 3728  df-ss 3735  df-nul 4062  df-sn 4315  df-uni 4573  df-iota 5994  df-fv 6039 This theorem is referenced by:  fvprc  6326  tz6.12i  6355  ndmfv  6359  nfunsn  6366  funpartfv  32383  setrec2lem1  42958
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