Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 18-Apr-2013)
Ref | Expression | ||
---|---|---|---|
Hypothesis | cdleme31fv2.f | |- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) ) |
|
Assertion | cdleme31id | |- ( ( X e. B /\ P = Q ) -> ( F ` X ) = X ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme31fv2.f | |- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) ) |
|
2 | simpl | |- ( ( P =/= Q /\ -. X .<_ W ) -> P =/= Q ) |
|
3 | 2 | necon2bi | |- ( P = Q -> -. ( P =/= Q /\ -. X .<_ W ) ) |
4 | 1 | cdleme31fv2 | |- ( ( X e. B /\ -. ( P =/= Q /\ -. X .<_ W ) ) -> ( F ` X ) = X ) |
5 | 3 4 | sylan2 | |- ( ( X e. B /\ P = Q ) -> ( F ` X ) = X ) |