Theorem indpi1 30082

Description: Preimage of the singleton { 1 } by the indicator function. See i1f1lem 23456. (Contributed by Thierry Arnoux, 21-Aug-2017.)

Assertion

Ref	Expression
indpi1	|- ( ( O e. V /\ A C_ O ) -> ( `' ( (𝟭 ` O ) ` A ) " { 1 } ) = A )

Proof of Theorem indpi1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ind1a 30081	. . . . 5 |- ( ( O e. V /\ A C_ O /\ x e. O ) -> ( ( ( (𝟭 ` O ) ` A ) ` x ) = 1 <-> x e. A ) )
2	1	3expia 1267	. . . 4 |- ( ( O e. V /\ A C_ O ) -> ( x e. O -> ( ( ( (𝟭 ` O ) ` A ) ` x ) = 1 <-> x e. A ) ) )
3	2	pm5.32d 671	. . 3 |- ( ( O e. V /\ A C_ O ) -> ( ( x e. O /\ ( ( (𝟭 ` O ) ` A ) ` x ) = 1 ) <-> ( x e. O /\ x e. A ) ) )
4		indf 30077	. . . 4 |- ( ( O e. V /\ A C_ O ) -> ( (𝟭 ` O ) ` A ) : O --> { 0 , 1 } )
5		ffn 6045	. . . 4 |- ( ( (𝟭 ` O ) ` A ) : O --> { 0 , 1 } -> ( (𝟭 ` O ) ` A ) Fn O )
6		fniniseg 6338	. . . 4 |- ( ( (𝟭 ` O ) ` A ) Fn O -> ( x e. ( `' ( (𝟭 ` O ) ` A ) " { 1 } ) <-> ( x e. O /\ ( ( (𝟭 ` O ) ` A ) ` x ) = 1 ) ) )
7	4, 5, 6	3syl 18	. . 3 |- ( ( O e. V /\ A C_ O ) -> ( x e. ( `' ( (𝟭 ` O ) ` A ) " { 1 } ) <-> ( x e. O /\ ( ( (𝟭 ` O ) ` A ) ` x ) = 1 ) ) )
8		ssel 3597	. . . . 5 |- ( A C_ O -> ( x e. A -> x e. O ) )
9	8	pm4.71rd 667	. . . 4 |- ( A C_ O -> ( x e. A <-> ( x e. O /\ x e. A ) ) )
10	9	adantl 482	. . 3 |- ( ( O e. V /\ A C_ O ) -> ( x e. A <-> ( x e. O /\ x e. A ) ) )
11	3, 7, 10	3bitr4d 300	. 2 |- ( ( O e. V /\ A C_ O ) -> ( x e. ( `' ( (𝟭 ` O ) ` A ) " { 1 } ) <-> x e. A ) )
12	11	eqrdv 2620	1 |- ( ( O e. V /\ A C_ O ) -> ( `' ( (𝟭 ` O ) ` A ) " { 1 } ) = A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   {csn 4177   {cpr 4179   `'ccnv 5113   "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-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  eulerpartlemgf  30441