Theorem nfunsn 5712

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 2114	. . . . . . 7 |- ( E! y A F y -> E* y A F y )
2		vex 2818	. . . . . . . . . 10 |- y e. _V
3	2	brres 5049	. . . . . . . . 9 |- ( x ( F |` { A } ) y <-> ( x F y /\ x e. { A } ) )
4		velsn 3711	. . . . . . . . . . 11 |- ( x e. { A } <-> x = A )
5		breq1 4117	. . . . . . . . . . 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 2148	. . . . . . 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 1923	. . . . 5 |- ( E! y A F y -> A. x E* y x ( F |` { A } ) y )
12		relres 5071	. . . . 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 5371	. . . 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 637	. 2 |- ( -. Fun ( F |` { A } ) -> -. E! y A F y )
17		tz6.12-2 5666	. 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 1396   = wceq 1398   E!weu 2082   E*wmo 2083   e. wcel 2205   (/)c0 3512   {csn 3694   class class class wbr 4114   |` cres 4756   Rel wrel 4759   Fun wfun 5351   ` cfv 5357

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-res 4766  df-iota 5317  df-fun 5359  df-fv 5365

This theorem is referenced by: (None)