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Theorem nfunsn 6187
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 2498 . . . . . . 7 (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝐴𝐹𝑦)
2 vex 3192 . . . . . . . . . 10 𝑦 ∈ V
32brres 5367 . . . . . . . . 9 (𝑥(𝐹 ↾ {𝐴})𝑦 ↔ (𝑥𝐹𝑦𝑥 ∈ {𝐴}))
4 velsn 4169 . . . . . . . . . . 11 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
5 breq1 4621 . . . . . . . . . . 11 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
64, 5sylbi 207 . . . . . . . . . 10 (𝑥 ∈ {𝐴} → (𝑥𝐹𝑦𝐴𝐹𝑦))
76biimpac 503 . . . . . . . . 9 ((𝑥𝐹𝑦𝑥 ∈ {𝐴}) → 𝐴𝐹𝑦)
83, 7sylbi 207 . . . . . . . 8 (𝑥(𝐹 ↾ {𝐴})𝑦𝐴𝐹𝑦)
98moimi 2519 . . . . . . 7 (∃*𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
101, 9syl 17 . . . . . 6 (∃!𝑦 𝐴𝐹𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
11 tz6.12-2 6144 . . . . . 6 (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹𝐴) = ∅)
1210, 11nsyl4 156 . . . . 5 (¬ (𝐹𝐴) = ∅ → ∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
1312alrimiv 1852 . . . 4 (¬ (𝐹𝐴) = ∅ → ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦)
14 relres 5390 . . . 4 Rel (𝐹 ↾ {𝐴})
1513, 14jctil 559 . . 3 (¬ (𝐹𝐴) = ∅ → (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
16 dffun6 5867 . . 3 (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ {𝐴})𝑦))
1715, 16sylibr 224 . 2 (¬ (𝐹𝐴) = ∅ → Fun (𝐹 ↾ {𝐴}))
1817con1i 144 1 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wcel 1987  ∃!weu 2469  ∃*wmo 2470  c0 3896  {csn 4153   class class class wbr 4618  cres 5081  Rel wrel 5084  Fun wfun 5846  cfv 5852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-res 5091  df-iota 5815  df-fun 5854  df-fv 5860
This theorem is referenced by:  fvfundmfvn0  6188  dffv2  6233
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