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Theorem nfunsn 5353
 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 2244 . . . . . 6 (∃!y AFy∃*y AFy)
2 brres 4949 . . . . . . . 8 (x(F {A})y ↔ (xFy x {A}))
3 elsn 3748 . . . . . . . . . 10 (x {A} ↔ x = A)
4 breq1 4642 . . . . . . . . . 10 (x = A → (xFyAFy))
53, 4sylbi 187 . . . . . . . . 9 (x {A} → (xFyAFy))
65biimpac 472 . . . . . . . 8 ((xFy x {A}) → AFy)
72, 6sylbi 187 . . . . . . 7 (x(F {A})yAFy)
87moimi 2251 . . . . . 6 (∃*y AFy∃*y x(F {A})y)
91, 8syl 15 . . . . 5 (∃!y AFy∃*y x(F {A})y)
10 tz6.12-2 5346 . . . . 5 ∃!y AFy → (FA) = )
119, 10nsyl4 134 . . . 4 (¬ (FA) = ∃*y x(F {A})y)
1211alrimiv 1631 . . 3 (¬ (FA) = x∃*y x(F {A})y)
13 dffun6 5124 . . 3 (Fun (F {A}) ↔ x∃*y x(F {A})y)
1412, 13sylibr 203 . 2 (¬ (FA) = → Fun (F {A}))
1514con1i 121 1 (¬ Fun (F {A}) → (FA) = )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  ∅c0 3550  {csn 3737   class class class wbr 4639   ↾ cres 4774  Fun wfun 4775   ‘cfv 4781 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-id 4767  df-xp 4784  df-cnv 4785  df-res 4788  df-fun 4789  df-fv 4795 This theorem is referenced by: (None)
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