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Theorem tz6.12-2 5419
 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 5138 . 2 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
2 iotanul 5110 . 2 (¬ ∃!𝑥 𝐴𝐹𝑥 → (℩𝑥𝐴𝐹𝑥) = ∅)
31, 2syl5eq 2185 1 (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1332  ∃!weu 2000  ∅c0 3367   class class class wbr 3936  ℩cio 5093  ‘cfv 5130 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3077  df-in 3081  df-ss 3088  df-nul 3368  df-sn 3537  df-uni 3744  df-iota 5095  df-fv 5138 This theorem is referenced by:  fvprc  5422  ndmfvg  5459  nfunsn  5462
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