Theorem indval 30075

Description: Value of the indicator function generator for a set A and a domain O. (Contributed by Thierry Arnoux, 2-Feb-2017.)

Assertion

Ref	Expression
indval	|- ( ( O e. V /\ A C_ O ) -> ( (𝟭 ` O ) ` A ) = ( x e. O |-> if ( x e. A , 1 , 0 ) ) )

Distinct variable groups:   x, O   x, A

Allowed substitution hint:   V( x)

Proof of Theorem indval

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		indv 30074	. . 3 |- ( O e. V -> (𝟭 ` O ) = ( a e. ~P O |-> ( x e. O |-> if ( x e. a , 1 , 0 ) ) ) )
2	1	adantr 481	. 2 |- ( ( O e. V /\ A C_ O ) -> (𝟭 ` O ) = ( a e. ~P O |-> ( x e. O |-> if ( x e. a , 1 , 0 ) ) ) )
3		eleq2 2690	. . . . 5 |- ( a = A -> ( x e. a <-> x e. A ) )
4	3	ifbid 4108	. . . 4 |- ( a = A -> if ( x e. a , 1 , 0 ) = if ( x e. A , 1 , 0 ) )
5	4	mpteq2dv 4745	. . 3 |- ( a = A -> ( x e. O |-> if ( x e. a , 1 , 0 ) ) = ( x e. O |-> if ( x e. A , 1 , 0 ) ) )
6	5	adantl 482	. 2 |- ( ( ( O e. V /\ A C_ O ) /\ a = A ) -> ( x e. O |-> if ( x e. a , 1 , 0 ) ) = ( x e. O |-> if ( x e. A , 1 , 0 ) ) )
7		simpr 477	. . 3 |- ( ( O e. V /\ A C_ O ) -> A C_ O )
8		ssexg 4804	. . . . 5 |- ( ( A C_ O /\ O e. V ) -> A e. _V )
9	8	ancoms 469	. . . 4 |- ( ( O e. V /\ A C_ O ) -> A e. _V )
10		elpwg 4166	. . . 4 |- ( A e. _V -> ( A e. ~P O <-> A C_ O ) )
11	9, 10	syl 17	. . 3 |- ( ( O e. V /\ A C_ O ) -> ( A e. ~P O <-> A C_ O ) )
12	7, 11	mpbird 247	. 2 |- ( ( O e. V /\ A C_ O ) -> A e. ~P O )
13		mptexg 6484	. . 3 |- ( O e. V -> ( x e. O |-> if ( x e. A , 1 , 0 ) ) e. _V )
14	13	adantr 481	. 2 |- ( ( O e. V /\ A C_ O ) -> ( x e. O |-> if ( x e. A , 1 , 0 ) ) e. _V )
15	2, 6, 12, 14	fvmptd 6288	1 |- ( ( O e. V /\ A C_ O ) -> ( (𝟭 ` O ) ` A ) = ( x e. O |-> if ( x e. A , 1 , 0 ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ifcif 4086   ~Pcpw 4158   |-> cmpt 4729   ` 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

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-ind 30073

This theorem is referenced by:  indval2  30076  indf  30077  indfval  30078