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Theorem tz6.12-2 5280
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 5010 . 2 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
2 iotanul 4982 . 2 (¬ ∃!𝑥 𝐴𝐹𝑥 → (℩𝑥𝐴𝐹𝑥) = ∅)
31, 2syl5eq 2132 1 (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1289  ∃!weu 1948  c0 3284   class class class wbr 3837  cio 4965  cfv 5002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-in 3003  df-ss 3010  df-nul 3285  df-sn 3447  df-uni 3649  df-iota 4967  df-fv 5010
This theorem is referenced by:  fvprc  5283  ndmfvg  5319  nfunsn  5322
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