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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 5498 |
. . . . . . . . . 10
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3 | 2 | eleq2d 2240 |
. . . . . . . . 9
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4 | 3 | anbi2d 461 |
. . . . . . . 8
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5 | 4 | rexbidv 2471 |
. . . . . . 7
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6 | 5 | orbi1d 786 |
. . . . . 6
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7 | 6 | abbidv 2288 |
. . . . 5
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8 | 7 | mpteq2dva 4079 |
. . . 4
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9 | recseq 6285 |
. . . 4
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10 | 8, 9 | syl 14 |
. . 3
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11 | 10 | reseq1d 4890 |
. 2
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12 | df-frec 6370 |
. 2
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13 | df-frec 6370 |
. 2
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14 | 11, 12, 13 | 3eqtr4g 2228 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-in 3127 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-res 4623 df-iota 5160 df-fv 5206 df-recs 6284 df-frec 6370 |
This theorem is referenced by: frecuzrdgdom 10374 frecuzrdgfun 10376 frecuzrdgsuct 10380 seqeq1 10404 seqeq2 10405 seqeq3 10406 iseqvalcbv 10413 hashfz1 10717 ennnfonelemr 12378 ctinfom 12383 isomninn 14063 iswomninn 14082 ismkvnn 14085 |
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