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Mirrors > Home > ILE Home > Th. List > iseqeq2 | Unicode version |
Description: Equality theorem for the
sequence builder operation.
New proofs should use seqeq2 9925 instead (together with iseqsst 9949 or iseqseq3 9965 if need be). (Contributed by Jim Kingdon, 30-May-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
iseqeq2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 944 |
. . . . . . 7
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2 | 1 | oveqd 5685 |
. . . . . 6
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3 | 2 | opeq2d 3637 |
. . . . 5
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4 | 3 | mpt2eq3dva 5729 |
. . . 4
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5 | freceq1 6173 |
. . . 4
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6 | 4, 5 | syl 14 |
. . 3
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7 | 6 | rneqd 4679 |
. 2
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8 | df-iseq 9916 |
. 2
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9 | df-iseq 9916 |
. 2
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10 | 7, 8, 9 | 3eqtr4g 2146 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2624 df-un 3006 df-in 3008 df-ss 3015 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-br 3854 df-opab 3908 df-mpt 3909 df-cnv 4462 df-dm 4464 df-rn 4465 df-res 4466 df-iota 4995 df-fv 5038 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-recs 6086 df-frec 6172 df-iseq 9916 |
This theorem is referenced by: seqeq2 9925 |
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