MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfunsn Structured version   Visualization version   GIF version

Theorem nfunsn 6943
Description: If the restriction of a class to a singleton is not a function, then its value is the empty set. (An artifact of our function value definition.) (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 2567 . . . . . . 7 (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝐴𝐹𝑦)
2 vex 3466 . . . . . . . . . 10 𝑦 ∈ V
32brresi 5998 . . . . . . . . 9 (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦))
4 velsn 4649 . . . . . . . . . . 11 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
5 breq1 5156 . . . . . . . . . . 11 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
64, 5sylbi 216 . . . . . . . . . 10 (𝑥 ∈ {𝐴} → (𝑥𝐹𝑦𝐴𝐹𝑦))
76biimpa 475 . . . . . . . . 9 ((𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) → 𝐴𝐹𝑦)
83, 7sylbi 216 . . . . . . . 8 (𝑥(𝐹 ↾ {𝐴})𝑦𝐴𝐹𝑦)
98moimi 2534 . . . . . . 7 (∃*𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
101, 9syl 17 . . . . . 6 (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
11 tz6.12-2 6889 . . . . . 6 (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹𝐴) = ∅)
1210, 11nsyl4 158 . . . . 5 (¬ (𝐹𝐴) = ∅ → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
1312alrimiv 1923 . . . 4 (¬ (𝐹𝐴) = ∅ → ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
14 relres 6015 . . . 4 Rel (𝐹 ↾ {𝐴})
1513, 14jctil 518 . . 3 (¬ (𝐹𝐴) = ∅ → (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
16 dffun6 6567 . . 3 (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
1715, 16sylibr 233 . 2 (¬ (𝐹𝐴) = ∅ → Fun (𝐹 ↾ {𝐴}))
1817con1i 147 1 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wal 1532   = wceq 1534  wcel 2099  ∃*wmo 2527  ∃!weu 2557  c0 4325  {csn 4633   class class class wbr 5153  cres 5684  Rel wrel 5687  Fun wfun 6548  cfv 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-res 5694  df-iota 6506  df-fun 6556  df-fv 6562
This theorem is referenced by:  fvfundmfvn0  6944  dffv2  6997  afv2ndeffv0  46873
  Copyright terms: Public domain W3C validator