Theorem invfval

Description: Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Hypotheses	invfval.b	|- B = ( Base ` C )
	invfval.n	|- N = ( Inv ` C )
	invfval.c	|- ( ph -> C e. Cat )
	invfval.x	|- ( ph -> X e. B )
	invfval.y	|- ( ph -> Y e. B )
	invfval.s	|- S = ( Sect ` C )
Assertion	invfval	|- ( ph -> ( X N Y ) = ( ( X S Y ) i^i `' ( Y S X ) ) )

Proof

Step	Hyp	Ref	Expression
1		invfval.b	|- B = ( Base ` C )
2		invfval.n	|- N = ( Inv ` C )
3		invfval.c	|- ( ph -> C e. Cat )
4		invfval.x	|- ( ph -> X e. B )
5		invfval.y	|- ( ph -> Y e. B )
6		invfval.s	|- S = ( Sect ` C )
7	1 2 3 4 4 6	invffval	|- ( ph -> N = ( x e. B , y e. B |-> ( ( x S y ) i^i `' ( y S x ) ) ) )
8		simprl	|- ( ( ph /\ ( x = X /\ y = Y ) ) -> x = X )
9		simprr	|- ( ( ph /\ ( x = X /\ y = Y ) ) -> y = Y )
10	8 9	oveq12d	|- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( x S y ) = ( X S Y ) )
11	9 8	oveq12d	|- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( y S x ) = ( Y S X ) )
12	11	cnveqd	|- ( ( ph /\ ( x = X /\ y = Y ) ) -> `' ( y S x ) = `' ( Y S X ) )
13	10 12	ineq12d	|- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( ( x S y ) i^i `' ( y S x ) ) = ( ( X S Y ) i^i `' ( Y S X ) ) )
14		ovex	|- ( X S Y ) e. _V
15	14	inex1	|- ( ( X S Y ) i^i `' ( Y S X ) ) e. _V
16	15	a1i	|- ( ph -> ( ( X S Y ) i^i `' ( Y S X ) ) e. _V )
17	7 13 4 5 16	ovmpod	|- ( ph -> ( X N Y ) = ( ( X S Y ) i^i `' ( Y S X ) ) )