Description: An upper bound on the rank of a function. (Contributed by Gérard Lang, 5-Aug-2018)
Ref | Expression | ||
---|---|---|---|
Hypotheses | rankxpl.1 | |- A e. _V |
|
rankxpl.2 | |- B e. _V |
||
Assertion | rankfu | |- ( F : A --> B -> ( rank ` F ) C_ suc suc ( rank ` ( A u. B ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankxpl.1 | |- A e. _V |
|
2 | rankxpl.2 | |- B e. _V |
|
3 | fssxp | |- ( F : A --> B -> F C_ ( A X. B ) ) |
|
4 | 1 2 | xpex | |- ( A X. B ) e. _V |
5 | 4 | rankss | |- ( F C_ ( A X. B ) -> ( rank ` F ) C_ ( rank ` ( A X. B ) ) ) |
6 | 1 2 | rankxpu | |- ( rank ` ( A X. B ) ) C_ suc suc ( rank ` ( A u. B ) ) |
7 | 5 6 | sstrdi | |- ( F C_ ( A X. B ) -> ( rank ` F ) C_ suc suc ( rank ` ( A u. B ) ) ) |
8 | 3 7 | syl | |- ( F : A --> B -> ( rank ` F ) C_ suc suc ( rank ` ( A u. B ) ) ) |