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}) → (F ‘A) = ∅)

Proof of Theorem nfunsn

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	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 → (xFy ↔ AFy))
5	3, 4	sylbi 187	. . . . . . . . 9 ⊢ (x ∈ {A} → (xFy ↔ AFy))
6	5	biimpac 472	. . . . . . . 8 ⊢ ((xFy ∧ x ∈ {A}) → AFy)
7	2, 6	sylbi 187	. . . . . . 7 ⊢ (x(F ↾ {A})y → AFy)
8	7	moimi 2251	. . . . . 6 ⊢ (∃*y AFy → ∃*y x(F ↾ {A})y)
9	1, 8	syl 15	. . . . 5 ⊢ (∃!y AFy → ∃*y x(F ↾ {A})y)
10		tz6.12-2 5346	. . . . 5 ⊢ (¬ ∃!y AFy → (F ‘A) = ∅)
11	9, 10	nsyl4 134	. . . 4 ⊢ (¬ (F ‘A) = ∅ → ∃*y x(F ↾ {A})y)
12	11	alrimiv 1631	. . 3 ⊢ (¬ (F ‘A) = ∅ → ∀x∃*y x(F ↾ {A})y)
13		dffun6 5124	. . 3 ⊢ (Fun (F ↾ {A}) ↔ ∀x∃*y x(F ↾ {A})y)
14	12, 13	sylibr 203	. 2 ⊢ (¬ (F ‘A) = ∅ → Fun (F ↾ {A}))
15	14	con1i 121	1 ⊢ (¬ Fun (F ↾ {A}) → (F ‘A) = ∅)

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-mp 5  ax-1 6  ax-2 7  ax-3 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)