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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 2164 | . . . 4 recs recs | |
2 | eqid 2164 | . . . . 5 | |
3 | 2 | frectfr 6359 | . . . 4 |
4 | 1, 3 | tfri1d 6294 | . . 3 recs |
5 | fnresin1 5296 | . . 3 recs recs | |
6 | 4, 5 | syl 14 | . 2 recs |
7 | omsson 4584 | . . . . . 6 | |
8 | sseqin2 3336 | . . . . . 6 | |
9 | 7, 8 | mpbi 144 | . . . . 5 |
10 | 9 | reseq2i 4875 | . . . 4 recs recs |
11 | df-frec 6350 | . . . 4 frec recs | |
12 | 10, 11 | eqtr4i 2188 | . . 3 recs frec |
13 | fneq12 5275 | . . 3 recs frec recs frec | |
14 | 12, 9, 13 | mp2an 423 | . 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 698 wal 1340 wceq 1342 wcel 2135 cab 2150 wrex 2443 cvv 2721 cin 3110 wss 3111 c0 3404 cmpt 4037 con0 4335 csuc 4337 com 4561 cdm 4598 cres 4600 wfn 5177 cfv 5182 recscrecs 6263 freccfrec 6349 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-recs 6264 df-frec 6350 |
This theorem is referenced by: frecrdg 6367 frec2uzrand 10330 frec2uzf1od 10331 frecfzennn 10351 hashinfom 10680 |
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