Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016)
Ref | Expression | ||
---|---|---|---|
Hypothesis | iunxsng.1 | |- ( x = A -> B = C ) |
|
Assertion | iunxsng | |- ( A e. V -> U_ x e. { A } B = C ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunxsng.1 | |- ( x = A -> B = C ) |
|
2 | eliun | |- ( y e. U_ x e. { A } B <-> E. x e. { A } y e. B ) |
|
3 | 1 | eleq2d | |- ( x = A -> ( y e. B <-> y e. C ) ) |
4 | 3 | rexsng | |- ( A e. V -> ( E. x e. { A } y e. B <-> y e. C ) ) |
5 | 2 4 | syl5bb | |- ( A e. V -> ( y e. U_ x e. { A } B <-> y e. C ) ) |
6 | 5 | eqrdv | |- ( A e. V -> U_ x e. { A } B = C ) |