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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 5355 |
. . . . . . . . . 10
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3 | 2 | eleq2d 2169 |
. . . . . . . . 9
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4 | 3 | anbi2d 455 |
. . . . . . . 8
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5 | 4 | rexbidv 2397 |
. . . . . . 7
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6 | 5 | orbi1d 746 |
. . . . . 6
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7 | 6 | abbidv 2217 |
. . . . 5
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8 | 7 | mpteq2dva 3958 |
. . . 4
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9 | recseq 6133 |
. . . 4
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10 | 8, 9 | syl 14 |
. . 3
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11 | 10 | reseq1d 4754 |
. 2
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12 | df-frec 6218 |
. 2
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13 | df-frec 6218 |
. 2
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14 | 11, 12, 13 | 3eqtr4g 2157 |
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 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-in 3027 df-uni 3684 df-br 3876 df-opab 3930 df-mpt 3931 df-res 4489 df-iota 5024 df-fv 5067 df-recs 6132 df-frec 6218 |
This theorem is referenced by: frecuzrdgdom 10032 frecuzrdgfun 10034 frecuzrdgsuct 10038 seqeq1 10062 seqeq2 10063 seqeq3 10064 iseqvalcbv 10071 hashfz1 10370 ennnfonelemr 11728 ctinfom 11733 isomninn 12810 |
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