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Theorem nfunsn 6135
 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 2487 . . . . . . 7 (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝐴𝐹𝑦)
2 vex 3176 . . . . . . . . . 10 𝑦 ∈ V
32brres 5323 . . . . . . . . 9 (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥𝐹𝑦𝑥 ∈ {𝐴}))
4 velsn 4141 . . . . . . . . . . 11 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
5 breq1 4586 . . . . . . . . . . 11 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
64, 5sylbi 206 . . . . . . . . . 10 (𝑥 ∈ {𝐴} → (𝑥𝐹𝑦𝐴𝐹𝑦))
76biimpac 502 . . . . . . . . 9 ((𝑥𝐹𝑦𝑥 ∈ {𝐴}) → 𝐴𝐹𝑦)
83, 7sylbi 206 . . . . . . . 8 (𝑥(𝐹 ↾ {𝐴})𝑦𝐴𝐹𝑦)
98moimi 2508 . . . . . . 7 (∃*𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
101, 9syl 17 . . . . . 6 (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
11 tz6.12-2 6094 . . . . . 6 (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹𝐴) = ∅)
1210, 11nsyl4 155 . . . . 5 (¬ (𝐹𝐴) = ∅ → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
1312alrimiv 1842 . . . 4 (¬ (𝐹𝐴) = ∅ → ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
14 relres 5346 . . . 4 Rel (𝐹 ↾ {𝐴})
1513, 14jctil 558 . . 3 (¬ (𝐹𝐴) = ∅ → (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
16 dffun6 5819 . . 3 (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
1715, 16sylibr 223 . 2 (¬ (𝐹𝐴) = ∅ → Fun (𝐹 ↾ {𝐴}))
1817con1i 143 1 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∃!weu 2458  ∃*wmo 2459  ∅c0 3874  {csn 4125   class class class wbr 4583   ↾ cres 5040  Rel wrel 5043  Fun wfun 5798  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812 This theorem is referenced by:  fvfundmfvn0  6136  dffv2  6181
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