Theorem nfunsn 4706

Description: If the restriction of a class to a singleton is not a function, its value is the empty set. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

Ref	Expression
nfunsn	|- (-. Fun (F |` {A}) -> (F` A) = (/))

Proof of Theorem nfunsn

Step	Hyp	Ref	Expression
1		eumo 1807	. . . . . . 7 |- (E!y AFy -> E*y AFy)
2		visset 2295	. . . . . . . . . 10 |- y e. _V
3	2	brres 4223	. . . . . . . . 9 |- (x(F |` {A})y <-> (xFy /\ x e. {A}))
4		elsn 3058	. . . . . . . . . . 11 |- (x e. {A} <-> x = A)
5		breq1 3341	. . . . . . . . . . 11 |- (x = A -> (xFy <-> AFy))
6	4, 5	sylbi 216	. . . . . . . . . 10 |- (x e. {A} -> (xFy <-> AFy))
7	6	biimpac 462	. . . . . . . . 9 |- ((xFy /\ x e. {A}) -> AFy)
8	3, 7	sylbi 216	. . . . . . . 8 |- (x(F |` {A})y -> AFy)
9	8	immoi 1814	. . . . . . 7 |- (E*y AFy -> E*y x(F |` {A})y)
10	1, 9	syl 12	. . . . . 6 |- (E!y AFy -> E*y x(F |` {A})y)
11		tz6.12-2 4696	. . . . . 6 |- (-. E!y AFy -> (F` A) = (/))
12	10, 11	nsyl4 135	. . . . 5 |- (-. (F` A) = (/) -> E*y x(F |` {A})y)
13	12	19.21aiv 1664	. . . 4 |- (-. (F` A) = (/) -> A.xE*y x(F |` {A})y)
14		relres 4242	. . . 4 |- Rel (F |` {A})
15	13, 14	jctil 316	. . 3 |- (-. (F` A) = (/) -> (Rel (F |` {A}) /\ A.xE*y x(F |` {A})y))
16		dffun6 4436	. . 3 |- (Fun (F |` {A}) <-> (Rel (F |` {A}) /\ A.xE*y x(F |` {A})y))
17	15, 16	sylibr 217	. 2 |- (-. (F` A) = (/) -> Fun (F |` {A}))
18	17	con1i 112	1 |- (-. Fun (F |` {A}) -> (F` A) = (/))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E!weu 1771  E*wmo 1772  (/)c0 2875  {csn 3044   class class class wbr 3338   |` cres 3988  Rel wrel 3991  Fun wfun 3992  ` cfv 3998

This theorem is referenced by:  dffv2 4734

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014

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