Theorem nfunsn 6225

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 2499	. . . . . . 7 |- ( E! y A F y -> E* y A F y )
2		vex 3203	. . . . . . . . . 10 |- y e. _V
3	2	brres 5402	. . . . . . . . 9 |- ( x ( F |` { A } ) y <-> ( x F y /\ x e. { A } ) )
4		velsn 4193	. . . . . . . . . . 11 |- ( x e. { A } <-> x = A )
5		breq1 4656	. . . . . . . . . . 11 |- ( x = A -> ( x F y <-> A F y ) )
6	4, 5	sylbi 207	. . . . . . . . . 10 |- ( x e. { A } -> ( x F y <-> A F y ) )
7	6	biimpac 503	. . . . . . . . 9 |- ( ( x F y /\ x e. { A } ) -> A F y )
8	3, 7	sylbi 207	. . . . . . . 8 |- ( x ( F |` { A } ) y -> A F y )
9	8	moimi 2520	. . . . . . 7 |- ( E* y A F y -> E* y x ( F |` { A } ) y )
10	1, 9	syl 17	. . . . . 6 |- ( E! y A F y -> E* y x ( F |` { A } ) y )
11		tz6.12-2 6182	. . . . . 6 |- ( -. E! y A F y -> ( F ` A ) = (/) )
12	10, 11	nsyl4 156	. . . . 5 |- ( -. ( F ` A ) = (/) -> E* y x ( F |` { A } ) y )
13	12	alrimiv 1855	. . . 4 |- ( -. ( F ` A ) = (/) -> A. x E* y x ( F |` { A } ) y )
14		relres 5426	. . . 4 |- Rel ( F |` { A } )
15	13, 14	jctil 560	. . 3 |- ( -. ( F ` A ) = (/) -> ( Rel ( F |` { A } ) /\ A. x E* y x ( F |` { A } ) y ) )
16		dffun6 5903	. . 3 |- ( Fun ( F |` { A } ) <-> ( Rel ( F |` { A } ) /\ A. x E* y x ( F |` { A } ) y ) )
17	15, 16	sylibr 224	. 2 |- ( -. ( F ` A ) = (/) -> Fun ( F |` { A } ) )
18	17	con1i 144	1 |- ( -. Fun ( F |` { A } ) -> ( F ` A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   E!weu 2470   E*wmo 2471   (/)c0 3915   {csn 4177   class class class wbr 4653   |` cres 5116   Rel wrel 5119   Fun wfun 5882   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  fvfundmfvn0  6226  dffv2  6271