Theorem nfunsn 5730

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 2118	. . . . . . 7 |- ( E! y A F y -> E* y A F y )
2		vex 2824	. . . . . . . . . 10 |- y e. _V
3	2	brres 5067	. . . . . . . . 9 |- ( x ( F |` { A } ) y <-> ( x F y /\ x e. { A } ) )
4		velsn 3725	. . . . . . . . . . 11 |- ( x e. { A } <-> x = A )
5		breq1 4131	. . . . . . . . . . 11 |- ( x = A -> ( x F y <-> A F y ) )
6	4, 5	sylbi 121	. . . . . . . . . 10 |- ( x e. { A } -> ( x F y <-> A F y ) )
7	6	biimpac 298	. . . . . . . . 9 |- ( ( x F y /\ x e. { A } ) -> A F y )
8	3, 7	sylbi 121	. . . . . . . 8 |- ( x ( F |` { A } ) y -> A F y )
9	8	moimi 2152	. . . . . . 7 |- ( E* y A F y -> E* y x ( F |` { A } ) y )
10	1, 9	syl 14	. . . . . 6 |- ( E! y A F y -> E* y x ( F |` { A } ) y )
11	10	alrimiv 1927	. . . . 5 |- ( E! y A F y -> A. x E* y x ( F |` { A } ) y )
12		relres 5089	. . . . 5 |- Rel ( F |` { A } )
13	11, 12	jctil 312	. . . 4 |- ( E! y A F y -> ( Rel ( F |` { A } ) /\ A. x E* y x ( F |` { A } ) y ) )
14		dffun6 5389	. . . 4 |- ( Fun ( F |` { A } ) <-> ( Rel ( F |` { A } ) /\ A. x E* y x ( F |` { A } ) y ) )
15	13, 14	sylibr 134	. . 3 |- ( E! y A F y -> Fun ( F |` { A } ) )
16	15	con3i 641	. 2 |- ( -. Fun ( F |` { A } ) -> -. E! y A F y )
17		tz6.12-2 5684	. 2 |- ( -. E! y A F y -> ( F ` A ) = (/) )
18	16, 17	syl 14	1 |- ( -. Fun ( F |` { A } ) -> ( F ` A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1400   = wceq 1402   E!weu 2086   E*wmo 2087   e. wcel 2209   (/)c0 3520   {csn 3708   class class class wbr 4128   |` cres 4774   Rel wrel 4777   Fun wfun 5369   ` cfv 5375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-res 4784  df-iota 5335  df-fun 5377  df-fv 5383

This theorem is referenced by: (None)