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Theorem nfunsn 5338
Description: If the restriction of a class to a singleton is not a function, its value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
nfunsn (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ∅)

Proof of Theorem nfunsn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eumo 1980 . . . . . . 7 (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝐴𝐹𝑦)
2 vex 2622 . . . . . . . . . 10 𝑦 ∈ V
32brres 4719 . . . . . . . . 9 (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥𝐹𝑦𝑥 ∈ {𝐴}))
4 velsn 3463 . . . . . . . . . . 11 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
5 breq1 3848 . . . . . . . . . . 11 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
64, 5sylbi 119 . . . . . . . . . 10 (𝑥 ∈ {𝐴} → (𝑥𝐹𝑦𝐴𝐹𝑦))
76biimpac 292 . . . . . . . . 9 ((𝑥𝐹𝑦𝑥 ∈ {𝐴}) → 𝐴𝐹𝑦)
83, 7sylbi 119 . . . . . . . 8 (𝑥(𝐹 ↾ {𝐴})𝑦𝐴𝐹𝑦)
98moimi 2013 . . . . . . 7 (∃*𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
101, 9syl 14 . . . . . 6 (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
1110alrimiv 1802 . . . . 5 (∃!𝑦 𝐴𝐹𝑦 → ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
12 relres 4741 . . . . 5 Rel (𝐹 ↾ {𝐴})
1311, 12jctil 305 . . . 4 (∃!𝑦 𝐴𝐹𝑦 → (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
14 dffun6 5029 . . . 4 (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
1513, 14sylibr 132 . . 3 (∃!𝑦 𝐴𝐹𝑦 → Fun (𝐹 ↾ {𝐴}))
1615con3i 597 . 2 (¬ Fun (𝐹 ↾ {𝐴}) → ¬ ∃!𝑦 𝐴𝐹𝑦)
17 tz6.12-2 5296 . 2 (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹𝐴) = ∅)
1816, 17syl 14 1 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wal 1287   = wceq 1289  wcel 1438  ∃!weu 1948  ∃*wmo 1949  c0 3286  {csn 3446   class class class wbr 3845  cres 4440  Rel wrel 4443  Fun wfun 5009  cfv 5015
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-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-res 4450  df-iota 4980  df-fun 5017  df-fv 5023
This theorem is referenced by: (None)
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