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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 | frec frec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . . . . . . . . 11 | |
2 | 1 | fveq1d 5391 | . . . . . . . . . 10 |
3 | 2 | eleq2d 2187 | . . . . . . . . 9 |
4 | 3 | anbi2d 459 | . . . . . . . 8 |
5 | 4 | rexbidv 2415 | . . . . . . 7 |
6 | 5 | orbi1d 765 | . . . . . 6 |
7 | 6 | abbidv 2235 | . . . . 5 |
8 | 7 | mpteq2dva 3988 | . . . 4 |
9 | recseq 6171 | . . . 4 recs recs | |
10 | 8, 9 | syl 14 | . . 3 recs recs |
11 | 10 | reseq1d 4788 | . 2 recs recs |
12 | df-frec 6256 | . 2 frec recs | |
13 | df-frec 6256 | . 2 frec recs | |
14 | 11, 12, 13 | 3eqtr4g 2175 | 1 frec frec |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 682 wceq 1316 wcel 1465 cab 2103 wrex 2394 cvv 2660 c0 3333 cmpt 3959 csuc 4257 com 4474 cdm 4509 cres 4511 cfv 5093 recscrecs 6169 freccfrec 6255 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-in 3047 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-res 4521 df-iota 5058 df-fv 5101 df-recs 6170 df-frec 6256 |
This theorem is referenced by: frecuzrdgdom 10159 frecuzrdgfun 10161 frecuzrdgsuct 10165 seqeq1 10189 seqeq2 10190 seqeq3 10191 iseqvalcbv 10198 hashfz1 10497 ennnfonelemr 11863 ctinfom 11868 isomninn 13153 |
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