Description: Subset inheritance for set exponentiation. (Contributed by Glauco Siliprandi, 3-Mar-2021)
Ref | Expression | ||
---|---|---|---|
Hypotheses | mapssbi.a | |
|
mapssbi.b | |
||
mapssbi.c | |
||
mapssbi.n | |
||
Assertion | mapssbi | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapssbi.a | |
|
2 | mapssbi.b | |
|
3 | mapssbi.c | |
|
4 | mapssbi.n | |
|
5 | 2 | adantr | |
6 | simpr | |
|
7 | mapss | |
|
8 | 5 6 7 | syl2anc | |
9 | 8 | ex | |
10 | simplr | |
|
11 | nssrex | |
|
12 | 11 | biimpi | |
13 | 12 | adantl | |
14 | fconst6g | |
|
15 | 14 | adantl | |
16 | elmapg | |
|
17 | 1 3 16 | syl2anc | |
18 | 17 | adantr | |
19 | 15 18 | mpbird | |
20 | 19 | 3adant3 | |
21 | 3 | adantr | |
22 | 2 | adantr | |
23 | 4 | adantr | |
24 | simpr | |
|
25 | 21 22 23 24 | snelmap | |
26 | 25 | adantlr | |
27 | simplr | |
|
28 | 26 27 | pm2.65da | |
29 | 28 | 3adant2 | |
30 | nelss | |
|
31 | 20 29 30 | syl2anc | |
32 | 31 | 3exp | |
33 | 32 | adantr | |
34 | 33 | rexlimdv | |
35 | 13 34 | mpd | |
36 | 35 | adantlr | |
37 | 10 36 | condan | |
38 | 37 | ex | |
39 | 9 38 | impbid | |