Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016) (Revised by Thierry Arnoux, 2-May-2020) Avoid ax-13 . (Revised by Gino Giotto, 19-May-2023)
Ref | Expression | ||
---|---|---|---|
Hypotheses | iunxsngf.1 | |- F/_ x C |
|
iunxsngf.2 | |- ( x = A -> B = C ) |
||
Assertion | iunxsngf | |- ( A e. V -> U_ x e. { A } B = C ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunxsngf.1 | |- F/_ x C |
|
2 | iunxsngf.2 | |- ( x = A -> B = C ) |
|
3 | eliun | |- ( y e. U_ x e. { A } B <-> E. x e. { A } y e. B ) |
|
4 | 1 | nfcri | |- F/ x y e. C |
5 | 2 | eleq2d | |- ( x = A -> ( y e. B <-> y e. C ) ) |
6 | 4 5 | rexsngf | |- ( A e. V -> ( E. x e. { A } y e. B <-> y e. C ) ) |
7 | 3 6 | syl5bb | |- ( A e. V -> ( y e. U_ x e. { A } B <-> y e. C ) ) |
8 | 7 | eqrdv | |- ( A e. V -> U_ x e. { A } B = C ) |