Theorem indpreima 30087

Description: A function with range { 0 , 1 } as an indicator of the preimage of { 1 }. (Contributed by Thierry Arnoux, 23-Aug-2017.)

Assertion

Ref	Expression
indpreima	|- ( ( O e. V /\ F : O --> { 0 , 1 } ) -> F = ( (𝟭 ` O ) ` ( `' F " { 1 } ) ) )

Proof of Theorem indpreima

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffn 6045	. . 3 |- ( F : O --> { 0 , 1 } -> F Fn O )
2	1	adantl 482	. 2 |- ( ( O e. V /\ F : O --> { 0 , 1 } ) -> F Fn O )
3		cnvimass 5485	. . . . 5 |- ( `' F " { 1 } ) C_ dom F
4		fdm 6051	. . . . . 6 |- ( F : O --> { 0 , 1 } -> dom F = O )
5	4	adantl 482	. . . . 5 |- ( ( O e. V /\ F : O --> { 0 , 1 } ) -> dom F = O )
6	3, 5	syl5sseq 3653	. . . 4 |- ( ( O e. V /\ F : O --> { 0 , 1 } ) -> ( `' F " { 1 } ) C_ O )
7		indf 30077	. . . 4 |- ( ( O e. V /\ ( `' F " { 1 } ) C_ O ) -> ( (𝟭 ` O ) ` ( `' F " { 1 } ) ) : O --> { 0 , 1 } )
8	6, 7	syldan 487	. . 3 |- ( ( O e. V /\ F : O --> { 0 , 1 } ) -> ( (𝟭 ` O ) ` ( `' F " { 1 } ) ) : O --> { 0 , 1 } )
9		ffn 6045	. . 3 |- ( ( (𝟭 ` O ) ` ( `' F " { 1 } ) ) : O --> { 0 , 1 } -> ( (𝟭 ` O ) ` ( `' F " { 1 } ) ) Fn O )
10	8, 9	syl 17	. 2 |- ( ( O e. V /\ F : O --> { 0 , 1 } ) -> ( (𝟭 ` O ) ` ( `' F " { 1 } ) ) Fn O )
11		simpr 477	. . . . 5 |- ( ( O e. V /\ F : O --> { 0 , 1 } ) -> F : O --> { 0 , 1 } )
12	11	ffvelrnda 6359	. . . 4 |- ( ( ( O e. V /\ F : O --> { 0 , 1 } ) /\ x e. O ) -> ( F ` x ) e. { 0 , 1 } )
13		prcom 4267	. . . 4 |- { 0 , 1 } = { 1 , 0 }
14	12, 13	syl6eleq 2711	. . 3 |- ( ( ( O e. V /\ F : O --> { 0 , 1 } ) /\ x e. O ) -> ( F ` x ) e. { 1 , 0 } )
15	8	ffvelrnda 6359	. . . 4 |- ( ( ( O e. V /\ F : O --> { 0 , 1 } ) /\ x e. O ) -> ( ( (𝟭 ` O ) ` ( `' F " { 1 } ) ) ` x ) e. { 0 , 1 } )
16	15, 13	syl6eleq 2711	. . 3 |- ( ( ( O e. V /\ F : O --> { 0 , 1 } ) /\ x e. O ) -> ( ( (𝟭 ` O ) ` ( `' F " { 1 } ) ) ` x ) e. { 1 , 0 } )
17		simpll 790	. . . . 5 |- ( ( ( O e. V /\ F : O --> { 0 , 1 } ) /\ x e. O ) -> O e. V )
18	6	adantr 481	. . . . 5 |- ( ( ( O e. V /\ F : O --> { 0 , 1 } ) /\ x e. O ) -> ( `' F " { 1 } ) C_ O )
19		simpr 477	. . . . 5 |- ( ( ( O e. V /\ F : O --> { 0 , 1 } ) /\ x e. O ) -> x e. O )
20		ind1a 30081	. . . . 5 |- ( ( O e. V /\ ( `' F " { 1 } ) C_ O /\ x e. O ) -> ( ( ( (𝟭 ` O ) ` ( `' F " { 1 } ) ) ` x ) = 1 <-> x e. ( `' F " { 1 } ) ) )
21	17, 18, 19, 20	syl3anc 1326	. . . 4 |- ( ( ( O e. V /\ F : O --> { 0 , 1 } ) /\ x e. O ) -> ( ( ( (𝟭 ` O ) ` ( `' F " { 1 } ) ) ` x ) = 1 <-> x e. ( `' F " { 1 } ) ) )
22		fniniseg 6338	. . . . . 6 |- ( F Fn O -> ( x e. ( `' F " { 1 } ) <-> ( x e. O /\ ( F ` x ) = 1 ) ) )
23	2, 22	syl 17	. . . . 5 |- ( ( O e. V /\ F : O --> { 0 , 1 } ) -> ( x e. ( `' F " { 1 } ) <-> ( x e. O /\ ( F ` x ) = 1 ) ) )
24	23	baibd 948	. . . 4 |- ( ( ( O e. V /\ F : O --> { 0 , 1 } ) /\ x e. O ) -> ( x e. ( `' F " { 1 } ) <-> ( F ` x ) = 1 ) )
25	21, 24	bitr2d 269	. . 3 |- ( ( ( O e. V /\ F : O --> { 0 , 1 } ) /\ x e. O ) -> ( ( F ` x ) = 1 <-> ( ( (𝟭 ` O ) ` ( `' F " { 1 } ) ) ` x ) = 1 ) )
26	14, 16, 25	elpreq 29360	. 2 |- ( ( ( O e. V /\ F : O --> { 0 , 1 } ) /\ x e. O ) -> ( F ` x ) = ( ( (𝟭 ` O ) ` ( `' F " { 1 } ) ) ` x ) )
27	2, 10, 26	eqfnfvd 6314	1 |- ( ( O e. V /\ F : O --> { 0 , 1 } ) -> F = ( (𝟭 ` O ) ` ( `' F " { 1 } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   {csn 4177   {cpr 4179   `'ccnv 5113   dom cdm 5114   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888   0cc0 9936   1c1 9937  𝟭cind 30072

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009

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-reu 2919  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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-ind 30073

This theorem is referenced by:  indf1ofs  30088