Theorem maprnin

Description: Restricting the range of the mapping operator. (Contributed by Thierry Arnoux, 30-Aug-2017)

	Ref	Expression
Hypotheses	maprnin.1	|- A e. _V
	maprnin.2	|- B e. _V
Assertion	maprnin	|- ( ( B i^i C ) ^m A ) = { f e. ( B ^m A ) | ran f C_ C }

Proof

Step	Hyp	Ref	Expression
1		maprnin.1	|- A e. _V
2		maprnin.2	|- B e. _V
3		ffn	|- ( f : A --> B -> f Fn A )
4		df-f	|- ( f : A --> C <-> ( f Fn A /\ ran f C_ C ) )
5	4	baibr	|- ( f Fn A -> ( ran f C_ C <-> f : A --> C ) )
6	3 5	syl	|- ( f : A --> B -> ( ran f C_ C <-> f : A --> C ) )
7	6	pm5.32i	|- ( ( f : A --> B /\ ran f C_ C ) <-> ( f : A --> B /\ f : A --> C ) )
8	2 1	elmap	|- ( f e. ( B ^m A ) <-> f : A --> B )
9	8	anbi1i	|- ( ( f e. ( B ^m A ) /\ ran f C_ C ) <-> ( f : A --> B /\ ran f C_ C ) )
10		fin	|- ( f : A --> ( B i^i C ) <-> ( f : A --> B /\ f : A --> C ) )
11	7 9 10	3bitr4ri	|- ( f : A --> ( B i^i C ) <-> ( f e. ( B ^m A ) /\ ran f C_ C ) )
12	11	abbii	|- { f | f : A --> ( B i^i C ) } = { f | ( f e. ( B ^m A ) /\ ran f C_ C ) }
13	2	inex1	|- ( B i^i C ) e. _V
14	13 1	mapval	|- ( ( B i^i C ) ^m A ) = { f | f : A --> ( B i^i C ) }
15		df-rab	|- { f e. ( B ^m A ) | ran f C_ C } = { f | ( f e. ( B ^m A ) /\ ran f C_ C ) }
16	12 14 15	3eqtr4i	|- ( ( B i^i C ) ^m A ) = { f e. ( B ^m A ) | ran f C_ C }