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Mirrors > Home > ILE Home > Th. List > iseqeq1 | Unicode version |
Description: Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.) |
Ref | Expression |
---|---|
iseqeq1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . . . . 6 | |
2 | fveq2 5178 | . . . . . 6 | |
3 | 1, 2 | opeq12d 3557 | . . . . 5 |
4 | freceq2 5980 | . . . . 5 frec frec | |
5 | 3, 4 | syl 14 | . . . 4 frec frec |
6 | fveq2 5178 | . . . . . 6 | |
7 | eqid 2040 | . . . . . 6 | |
8 | mpt2eq12 5565 | . . . . . 6 | |
9 | 6, 7, 8 | sylancl 392 | . . . . 5 |
10 | freceq1 5979 | . . . . 5 frec frec | |
11 | 9, 10 | syl 14 | . . . 4 frec frec |
12 | 5, 11 | eqtrd 2072 | . . 3 frec frec |
13 | 12 | rneqd 4563 | . 2 frec frec |
14 | df-iseq 9212 | . 2 frec | |
15 | df-iseq 9212 | . 2 frec | |
16 | 13, 14, 15 | 3eqtr4g 2097 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1243 cop 3378 crn 4346 cfv 4902 (class class class)co 5512 cmpt2 5514 freccfrec 5977 c1 6890 caddc 6892 cuz 8473 cseq 9211 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-iota 4867 df-fv 4910 df-oprab 5516 df-mpt2 5517 df-recs 5920 df-frec 5978 df-iseq 9212 |
This theorem is referenced by: iseqid 9247 iiserex 9859 |
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