Description: The reflexive, transitive closure is indeed reflexive. (Contributed by Drahflow, 12-Nov-2015) (Revised by RP, 30-May-2020) (Revised by AV, 13-Jul-2024)
Ref | Expression | ||
---|---|---|---|
Hypotheses | rtrclreclem2.1 | |
|
rtrclreclem2.2 | |
||
Assertion | rtrclreclem2 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rtrclreclem2.1 | |
|
2 | rtrclreclem2.2 | |
|
3 | 0nn0 | |
|
4 | ssid | |
|
5 | 1 2 | relexp0d | |
6 | 4 5 | sseqtrrid | |
7 | oveq2 | |
|
8 | 7 | sseq2d | |
9 | 8 | rspcev | |
10 | 3 6 9 | sylancr | |
11 | ssiun | |
|
12 | 10 11 | syl | |
13 | 2 | elexd | |
14 | nn0ex | |
|
15 | ovex | |
|
16 | 14 15 | iunex | |
17 | oveq1 | |
|
18 | 17 | iuneq2d | |
19 | eqid | |
|
20 | 18 19 | fvmptg | |
21 | 13 16 20 | sylancl | |
22 | 12 21 | sseqtrrd | |
23 | df-rtrclrec | |
|
24 | fveq1 | |
|
25 | 24 | sseq2d | |
26 | 25 | imbi2d | |
27 | 23 26 | ax-mp | |
28 | 22 27 | mpbir | |