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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 5198 | . . . . . 6 | |
3 | 1, 2 | opeq12d 3578 | . . . . 5 |
4 | freceq2 6003 | . . . . 5 frec frec | |
5 | 3, 4 | syl 14 | . . . 4 frec frec |
6 | fveq2 5198 | . . . . . 6 | |
7 | eqid 2081 | . . . . . 6 | |
8 | mpt2eq12 5585 | . . . . . 6 | |
9 | 6, 7, 8 | sylancl 404 | . . . . 5 |
10 | freceq1 6002 | . . . . 5 frec frec | |
11 | 9, 10 | syl 14 | . . . 4 frec frec |
12 | 5, 11 | eqtrd 2113 | . . 3 frec frec |
13 | 12 | rneqd 4581 | . 2 frec frec |
14 | df-iseq 9432 | . 2 frec | |
15 | df-iseq 9432 | . 2 frec | |
16 | 13, 14, 15 | 3eqtr4g 2138 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1284 cop 3401 crn 4364 cfv 4922 (class class class)co 5532 cmpt2 5534 freccfrec 6000 c1 6982 caddc 6984 cuz 8619 cseq 9431 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-cnv 4371 df-dm 4373 df-rn 4374 df-res 4375 df-iota 4887 df-fv 4930 df-oprab 5536 df-mpt2 5537 df-recs 5943 df-frec 6001 df-iseq 9432 |
This theorem is referenced by: iseqid 9467 iseqz 9469 ibcval5 9690 bcn2 9691 iiserex 10177 |
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