Description: dist is unaffected by restriction. (Contributed by Mario Carneiro, 26-Aug-2015)
Ref | Expression | ||
---|---|---|---|
Hypotheses | ressds.1 | |- H = ( G |`s A ) |
|
ressds.2 | |- D = ( dist ` G ) |
||
Assertion | ressds | |- ( A e. V -> D = ( dist ` H ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressds.1 | |- H = ( G |`s A ) |
|
2 | ressds.2 | |- D = ( dist ` G ) |
|
3 | df-ds | |- dist = Slot ; 1 2 |
|
4 | 1nn0 | |- 1 e. NN0 |
|
5 | 2nn | |- 2 e. NN |
|
6 | 4 5 | decnncl | |- ; 1 2 e. NN |
7 | 1nn | |- 1 e. NN |
|
8 | 2nn0 | |- 2 e. NN0 |
|
9 | 1lt10 | |- 1 < ; 1 0 |
|
10 | 7 8 4 9 | declti | |- 1 < ; 1 2 |
11 | 1 2 3 6 10 | resslem | |- ( A e. V -> D = ( dist ` H ) ) |