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Theorem tz6.12-2 6653
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 6356 . 2 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
2 iotanul 6326 . 2 (¬ ∃!𝑥 𝐴𝐹𝑥 → (℩𝑥𝐴𝐹𝑥) = ∅)
31, 2syl5eq 2865 1 (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1528  ∃!weu 2646  c0 4288   class class class wbr 5057  cio 6305  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-v 3494  df-dif 3936  df-in 3940  df-ss 3949  df-nul 4289  df-sn 4558  df-uni 4831  df-iota 6307  df-fv 6356
This theorem is referenced by:  fvprc  6656  tz6.12i  6689  ndmfv  6693  nfunsn  6700  funpartfv  33303  setrec2lem1  44724
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