Theorem nfunsnOLD 4707

Description: If the restriction of a class to a singleton is not a function, its value is the empty set.

Assertion

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

Proof of Theorem nfunsnOLD

Step	Hyp	Ref	Expression
1		eleq2 1958	. . . . . . . . . . . . . . . 16 |- (dom ( F |` {A}) = {A} -> (x e. dom ( F |` {A}) <-> x e. {A}))
2		elsn 3058	. . . . . . . . . . . . . . . 16 |- (x e. {A} <-> x = A)
3	1, 2	syl6bb 595	. . . . . . . . . . . . . . 15 |- (dom ( F |` {A}) = {A} -> (x e. dom ( F |` {A}) <-> x = A))
4	3	biimpa 460	. . . . . . . . . . . . . 14 |- ((dom ( F |` {A}) = {A} /\ x e. dom ( F |` {A})) -> x = A)
5		snssi 3129	. . . . . . . . . . . . . . 15 |- (A e. dom ( F |` {A}) -> {A} C_ dom ( F |` {A}))
6		ssdmres 4235	. . . . . . . . . . . . . . . . 17 |- ({A} C_ dom ( F |` {A}) <-> dom ((F |` {A}) |` {A}) = {A})
7	6	biimpi 168	. . . . . . . . . . . . . . . 16 |- ({A} C_ dom ( F |` {A}) -> dom ((F |` {A}) |` {A}) = {A})
8		residm 4246	. . . . . . . . . . . . . . . . 17 |- ((F |` {A}) |` {A}) = (F |` {A})
9	8	dmeqi 4158	. . . . . . . . . . . . . . . 16 |- dom ((F |` {A}) |` {A}) = dom ( F |` {A})
10	7, 9	syl5eqr 1942	. . . . . . . . . . . . . . 15 |- ({A} C_ dom ( F |` {A}) -> dom ( F |` {A}) = {A})
11	5, 10	syl 12	. . . . . . . . . . . . . 14 |- (A e. dom ( F |` {A}) -> dom ( F |` {A}) = {A})
12	4, 11	sylan 497	. . . . . . . . . . . . 13 |- ((A e. dom ( F |` {A}) /\ x e. dom ( F |` {A})) -> x = A)
13		breq1 3341	. . . . . . . . . . . . . . 15 |- (x = A -> (x(F |` {A})y <-> A(F |` {A})y))
14	13	eubidv 1779	. . . . . . . . . . . . . 14 |- (x = A -> (E!y x(F |` {A})y <-> E!y A(F |` {A})y))
15	14	biimprd 171	. . . . . . . . . . . . 13 |- (x = A -> (E!y A(F |` {A})y -> E!y x(F |` {A})y))
16	12, 15	syl 12	. . . . . . . . . . . 12 |- ((A e. dom ( F |` {A}) /\ x e. dom ( F |` {A})) -> (E!y A(F |` {A})y -> E!y x(F |` {A})y))
17	16	ex 402	. . . . . . . . . . 11 |- (A e. dom ( F |` {A}) -> (x e. dom ( F |` {A}) -> (E!y A(F |` {A})y -> E!y x(F |` {A})y)))
18	17	com23 36	. . . . . . . . . 10 |- (A e. dom ( F |` {A}) -> (E!y A(F |` {A})y -> (x e. dom ( F |` {A}) -> E!y x(F |` {A})y)))
19	18	imp 377	. . . . . . . . 9 |- ((A e. dom ( F |` {A}) /\ E!y A(F |` {A})y) -> (x e. dom ( F |` {A}) -> E!y x(F |` {A})y))
20	19	r19.21aiv 2175	. . . . . . . 8 |- ((A e. dom ( F |` {A}) /\ E!y A(F |` {A})y) -> A.x e. dom ( F |` {A})E!y x(F |` {A})y)
21		relres 4242	. . . . . . . 8 |- Rel (F |` {A})
22	20, 21	jctil 316	. . . . . . 7 |- ((A e. dom ( F |` {A}) /\ E!y A(F |` {A})y) -> (Rel (F |` {A}) /\ A.x e. dom ( F |` {A})E!y x(F |` {A})y))
23	22	ex 402	. . . . . 6 |- (A e. dom ( F |` {A}) -> (E!y A(F |` {A})y -> (Rel (F |` {A}) /\ A.x e. dom ( F |` {A})E!y x(F |` {A})y)))
24		dffun8 4448	. . . . . 6 |- (Fun (F |` {A}) <-> (Rel (F |` {A}) /\ A.x e. dom ( F |` {A})E!y x(F |` {A})y))
25	23, 24	syl6ibr 230	. . . . 5 |- (A e. dom ( F |` {A}) -> (E!y A(F |` {A})y -> Fun (F |` {A})))
26	25	con3d 111	. . . 4 |- (A e. dom ( F |` {A}) -> (-. Fun (F |` {A}) -> -. E!y A(F |` {A})y))
27		tz6.12-2 4696	. . . 4 |- (-. E!y A(F |` {A})y -> ((F |` {A})` A) = (/))
28	26, 27	syl6com 64	. . 3 |- (-. Fun (F |` {A}) -> (A e. dom ( F |` {A}) -> ((F |` {A})` A) = (/)))
29		ndmfv 4702	. . 3 |- (-. A e. dom ( F |` {A}) -> ((F |` {A})` A) = (/))
30	28, 29	pm2.61d1 142	. 2 |- (-. Fun (F |` {A}) -> ((F |` {A})` A) = (/))
31		snidb 3068	. . . 4 |- (A e. _V <-> A e. {A})
32		fvres 4691	. . . 4 |- (A e. {A} -> ((F |` {A})` A) = (F` A))
33	31, 32	sylbi 216	. . 3 |- (A e. _V -> ((F |` {A})` A) = (F` A))
34		fvprc 4678	. . . 4 |- (-. A e. _V -> ((F |` {A})` A) = (/))
35		fvprc 4678	. . . 4 |- (-. A e. _V -> (F` A) = (/))
36	34, 35	eqtr4d 1928	. . 3 |- (-. A e. _V -> ((F |` {A})` A) = (F` A))
37	33, 36	pm2.61i 140	. 2 |- ((F |` {A})` A) = (F` A)
38	30, 37	syl5eqr 1942	1 |- (-. Fun (F |` {A}) -> (F` A) = (/))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E!weu 1771  A.wral 2105  _Vcvv 2292   C_ wss 2593  (/)c0 2875  {csn 3044   class class class wbr 3338  dom cdm 3986   |` cres 3988  Rel wrel 3991  Fun wfun 3992  ` cfv 3998

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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