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Theorem nfunsn 6926
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 2566 . . . . . . 7 (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝐴𝐹𝑦)
2 vex 3472 . . . . . . . . . 10 𝑦 ∈ V
32brresi 5983 . . . . . . . . 9 (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦))
4 velsn 4639 . . . . . . . . . . 11 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
5 breq1 5144 . . . . . . . . . . 11 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
64, 5sylbi 216 . . . . . . . . . 10 (𝑥 ∈ {𝐴} → (𝑥𝐹𝑦𝐴𝐹𝑦))
76biimpa 476 . . . . . . . . 9 ((𝑥 ∈ {𝐴} ∧ 𝑥𝐹𝑦) → 𝐴𝐹𝑦)
83, 7sylbi 216 . . . . . . . 8 (𝑥(𝐹 ↾ {𝐴})𝑦𝐴𝐹𝑦)
98moimi 2533 . . . . . . 7 (∃*𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
101, 9syl 17 . . . . . 6 (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
11 tz6.12-2 6872 . . . . . 6 (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹𝐴) = ∅)
1210, 11nsyl4 158 . . . . 5 (¬ (𝐹𝐴) = ∅ → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
1312alrimiv 1922 . . . 4 (¬ (𝐹𝐴) = ∅ → ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
14 relres 6003 . . . 4 Rel (𝐹 ↾ {𝐴})
1513, 14jctil 519 . . 3 (¬ (𝐹𝐴) = ∅ → (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
16 dffun6 6549 . . 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 395  wal 1531   = wceq 1533  wcel 2098  ∃*wmo 2526  ∃!weu 2556  c0 4317  {csn 4623   class class class wbr 5141  cres 5671  Rel wrel 5674  Fun wfun 6530  cfv 6536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-res 5681  df-iota 6488  df-fun 6538  df-fv 6544
This theorem is referenced by:  fvfundmfvn0  6927  dffv2  6979  afv2ndeffv0  46521
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