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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 |
. . . . . 6
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2 | fveq2 5178 |
. . . . . 6
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3 | 1, 2 | opeq12d 3557 |
. . . . 5
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4 | freceq2 5980 |
. . . . 5
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5 | 3, 4 | syl 14 |
. . . 4
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6 | fveq2 5178 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | eqid 2040 |
. . . . . 6
![]() ![]() ![]() ![]() | |
8 | mpt2eq12 5565 |
. . . . . 6
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9 | 6, 7, 8 | sylancl 392 |
. . . . 5
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10 | freceq1 5979 |
. . . . 5
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11 | 9, 10 | syl 14 |
. . . 4
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12 | 5, 11 | eqtrd 2072 |
. . 3
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13 | 12 | rneqd 4563 |
. 2
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14 | df-iseq 9212 |
. 2
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15 | df-iseq 9212 |
. 2
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16 | 13, 14, 15 | 3eqtr4g 2097 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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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