Theorem fnnfpeq0 6444

Description: A function is the identity iff it moves no points. (Contributed by Stefan O'Rear, 25-Aug-2015.)

Assertion

Ref	Expression
fnnfpeq0	|- ( F Fn A -> ( dom ( F \ _I ) = (/) <-> F = ( _I |` A ) ) )

Proof of Theorem fnnfpeq0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabeq0 3957	. . 3 |- ( { x e. A | ( F ` x ) =/= x } = (/) <-> A. x e. A -. ( F ` x ) =/= x )
2		fvresi 6439	. . . . . . 7 |- ( x e. A -> ( ( _I |` A ) ` x ) = x )
3	2	eqeq2d 2632	. . . . . 6 |- ( x e. A -> ( ( F ` x ) = ( ( _I |` A ) ` x ) <-> ( F ` x ) = x ) )
4	3	adantl 482	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) = ( ( _I |` A ) ` x ) <-> ( F ` x ) = x ) )
5		nne 2798	. . . . 5 |- ( -. ( F ` x ) =/= x <-> ( F ` x ) = x )
6	4, 5	syl6rbbr 279	. . . 4 |- ( ( F Fn A /\ x e. A ) -> ( -. ( F ` x ) =/= x <-> ( F ` x ) = ( ( _I |` A ) ` x ) ) )
7	6	ralbidva 2985	. . 3 |- ( F Fn A -> ( A. x e. A -. ( F ` x ) =/= x <-> A. x e. A ( F ` x ) = ( ( _I |` A ) ` x ) ) )
8	1, 7	syl5bb 272	. 2 |- ( F Fn A -> ( { x e. A | ( F ` x ) =/= x } = (/) <-> A. x e. A ( F ` x ) = ( ( _I |` A ) ` x ) ) )
9		fndifnfp 6442	. . 3 |- ( F Fn A -> dom ( F \ _I ) = { x e. A | ( F ` x ) =/= x } )
10	9	eqeq1d 2624	. 2 |- ( F Fn A -> ( dom ( F \ _I ) = (/) <-> { x e. A | ( F ` x ) =/= x } = (/) ) )
11		fnresi 6008	. . 3 |- ( _I |` A ) Fn A
12		eqfnfv 6311	. . 3 |- ( ( F Fn A /\ ( _I |` A ) Fn A ) -> ( F = ( _I |` A ) <-> A. x e. A ( F ` x ) = ( ( _I |` A ) ` x ) ) )
13	11, 12	mpan2 707	. 2 |- ( F Fn A -> ( F = ( _I |` A ) <-> A. x e. A ( F ` x ) = ( ( _I |` A ) ` x ) ) )
14	8, 10, 13	3bitr4d 300	1 |- ( F Fn A -> ( dom ( F \ _I ) = (/) <-> F = ( _I |` A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   \ cdif 3571   (/)c0 3915   _I cid 5023   dom cdm 5114   |` cres 5116   Fn wfn 5883   ` 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-8 1992  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-pow 4843  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-csb 3534  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-mpt 4730  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-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  symggen  17890  m1detdiag  20403  mdetdiaglem  20404