Theorem fndifnfp 6442

Description: Express the class of non-fixed points of a function. (Contributed by Stefan O'Rear, 14-Aug-2015.)

Assertion

Ref	Expression
fndifnfp	|- ( F Fn A -> dom ( F \ _I ) = { x e. A | ( F ` x ) =/= x } )

Distinct variable groups:   x, F   x, A

Proof of Theorem fndifnfp

Step	Hyp	Ref	Expression
1		dffn2 6047	. . . . . . . 8 |- ( F Fn A <-> F : A --> _V )
2		fssxp 6060	. . . . . . . 8 |- ( F : A --> _V -> F C_ ( A X. _V ) )
3	1, 2	sylbi 207	. . . . . . 7 |- ( F Fn A -> F C_ ( A X. _V ) )
4		ssdif0 3942	. . . . . . 7 |- ( F C_ ( A X. _V ) <-> ( F \ ( A X. _V ) ) = (/) )
5	3, 4	sylib 208	. . . . . 6 |- ( F Fn A -> ( F \ ( A X. _V ) ) = (/) )
6	5	uneq2d 3767	. . . . 5 |- ( F Fn A -> ( ( F \ _I ) u. ( F \ ( A X. _V ) ) ) = ( ( F \ _I ) u. (/) ) )
7		un0 3967	. . . . 5 |- ( ( F \ _I ) u. (/) ) = ( F \ _I )
8	6, 7	syl6req 2673	. . . 4 |- ( F Fn A -> ( F \ _I ) = ( ( F \ _I ) u. ( F \ ( A X. _V ) ) ) )
9		df-res 5126	. . . . . 6 |- ( _I |` A ) = ( _I i^i ( A X. _V ) )
10	9	difeq2i 3725	. . . . 5 |- ( F \ ( _I |` A ) ) = ( F \ ( _I i^i ( A X. _V ) ) )
11		difindi 3881	. . . . 5 |- ( F \ ( _I i^i ( A X. _V ) ) ) = ( ( F \ _I ) u. ( F \ ( A X. _V ) ) )
12	10, 11	eqtri 2644	. . . 4 |- ( F \ ( _I |` A ) ) = ( ( F \ _I ) u. ( F \ ( A X. _V ) ) )
13	8, 12	syl6eqr 2674	. . 3 |- ( F Fn A -> ( F \ _I ) = ( F \ ( _I |` A ) ) )
14	13	dmeqd 5326	. 2 |- ( F Fn A -> dom ( F \ _I ) = dom ( F \ ( _I |` A ) ) )
15		fnresi 6008	. . 3 |- ( _I |` A ) Fn A
16		fndmdif 6321	. . 3 |- ( ( F Fn A /\ ( _I |` A ) Fn A ) -> dom ( F \ ( _I |` A ) ) = { x e. A | ( F ` x ) =/= ( ( _I |` A ) ` x ) } )
17	15, 16	mpan2 707	. 2 |- ( F Fn A -> dom ( F \ ( _I |` A ) ) = { x e. A | ( F ` x ) =/= ( ( _I |` A ) ` x ) } )
18		fvresi 6439	. . . . 5 |- ( x e. A -> ( ( _I |` A ) ` x ) = x )
19	18	neeq2d 2854	. . . 4 |- ( x e. A -> ( ( F ` x ) =/= ( ( _I |` A ) ` x ) <-> ( F ` x ) =/= x ) )
20	19	rabbiia 3185	. . 3 |- { x e. A | ( F ` x ) =/= ( ( _I |` A ) ` x ) } = { x e. A | ( F ` x ) =/= x }
21	20	a1i 11	. 2 |- ( F Fn A -> { x e. A | ( F ` x ) =/= ( ( _I |` A ) ` x ) } = { x e. A | ( F ` x ) =/= x } )
22	14, 17, 21	3eqtrd 2660	1 |- ( F Fn A -> dom ( F \ _I ) = { x e. A | ( F ` x ) =/= x } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   _I cid 5023   X. cxp 5112   dom cdm 5114   |` cres 5116   Fn wfn 5883   -->wf 5884   ` 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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  fnelnfp  6443  fnnfpeq0  6444  f1omvdcnv  17864  pmtrmvd  17876  pmtrdifellem4  17899  sygbasnfpfi  17932