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Mirrors > Home > ILE Home > Th. List > freceq1 | Unicode version |
Description: Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
Ref | Expression |
---|---|
freceq1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 |
. . . . . . . . . . 11
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2 | 1 | fveq1d 5431 |
. . . . . . . . . 10
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3 | 2 | eleq2d 2210 |
. . . . . . . . 9
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4 | 3 | anbi2d 460 |
. . . . . . . 8
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5 | 4 | rexbidv 2439 |
. . . . . . 7
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6 | 5 | orbi1d 781 |
. . . . . 6
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7 | 6 | abbidv 2258 |
. . . . 5
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8 | 7 | mpteq2dva 4026 |
. . . 4
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9 | recseq 6211 |
. . . 4
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10 | 8, 9 | syl 14 |
. . 3
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11 | 10 | reseq1d 4826 |
. 2
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12 | df-frec 6296 |
. 2
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13 | df-frec 6296 |
. 2
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14 | 11, 12, 13 | 3eqtr4g 2198 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-in 3082 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-res 4559 df-iota 5096 df-fv 5139 df-recs 6210 df-frec 6296 |
This theorem is referenced by: frecuzrdgdom 10222 frecuzrdgfun 10224 frecuzrdgsuct 10228 seqeq1 10252 seqeq2 10253 seqeq3 10254 iseqvalcbv 10261 hashfz1 10561 ennnfonelemr 11972 ctinfom 11977 isomninn 13401 iswomninn 13418 ismkvnn 13420 |
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