Theorem nfunsn 5354

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 4950	. . . . . . . 8 ⊢ (x(F ↾ {A})y ↔ (xFy ∧ x ∈ {A}))
3		elsn 3749	. . . . . . . . . 10 ⊢ (x ∈ {A} ↔ x = A)
4		breq1 4643	. . . . . . . . . 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 5347	. . . . 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 5125	. . 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) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  ∅c0 3551  {csn 3738   class class class wbr 4640   ↾ cres 4775  Fun wfun 4776   ‘cfv 4782

This proof depends on 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof 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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-id 4768  df-xp 4785  df-cnv 4786  df-res 4789  df-fun 4790  df-fv 4796

This theorem is used by: (None)