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Mirrors > Home > ILE Home > Th. List > frecfnom | Unicode version |
Description: The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 13-May-2020.) |
Ref | Expression |
---|---|
frecfnom | frec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2137 | . . . 4 recs recs | |
2 | eqid 2137 | . . . . 5 | |
3 | 2 | frectfr 6290 | . . . 4 |
4 | 1, 3 | tfri1d 6225 | . . 3 recs |
5 | fnresin1 5232 | . . 3 recs recs | |
6 | 4, 5 | syl 14 | . 2 recs |
7 | omsson 4521 | . . . . . 6 | |
8 | sseqin2 3290 | . . . . . 6 | |
9 | 7, 8 | mpbi 144 | . . . . 5 |
10 | 9 | reseq2i 4811 | . . . 4 recs recs |
11 | df-frec 6281 | . . . 4 frec recs | |
12 | 10, 11 | eqtr4i 2161 | . . 3 recs frec |
13 | fneq12 5211 | . . 3 recs frec recs frec | |
14 | 12, 9, 13 | mp2an 422 | . 2 recs frec |
15 | 6, 14 | sylib 121 | 1 frec |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 697 wal 1329 wceq 1331 wcel 1480 cab 2123 wrex 2415 cvv 2681 cin 3065 wss 3066 c0 3358 cmpt 3984 con0 4280 csuc 4282 com 4499 cdm 4534 cres 4536 wfn 5113 cfv 5118 recscrecs 6194 freccfrec 6280 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-recs 6195 df-frec 6281 |
This theorem is referenced by: frecrdg 6298 frec2uzrand 10171 frec2uzf1od 10172 frecfzennn 10192 hashinfom 10517 |
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