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Mirrors > Home > ILE Home > Th. List > frec0g | Unicode version |
Description: The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.) |
Ref | Expression |
---|---|
frec0g | frec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dm0 4723 | . . . . . . . . . 10 | |
2 | 1 | biantrur 301 | . . . . . . . . 9 |
3 | vex 2663 | . . . . . . . . . . . . . . . 16 | |
4 | nsuceq0g 4310 | . . . . . . . . . . . . . . . 16 | |
5 | 3, 4 | ax-mp 5 | . . . . . . . . . . . . . . 15 |
6 | 5 | nesymi 2331 | . . . . . . . . . . . . . 14 |
7 | 1 | eqeq1i 2125 | . . . . . . . . . . . . . 14 |
8 | 6, 7 | mtbir 645 | . . . . . . . . . . . . 13 |
9 | 8 | intnanr 900 | . . . . . . . . . . . 12 |
10 | 9 | a1i 9 | . . . . . . . . . . 11 |
11 | 10 | nrex 2501 | . . . . . . . . . 10 |
12 | 11 | biorfi 720 | . . . . . . . . 9 |
13 | orcom 702 | . . . . . . . . 9 | |
14 | 2, 12, 13 | 3bitri 205 | . . . . . . . 8 |
15 | 14 | abbii 2233 | . . . . . . 7 |
16 | abid2 2238 | . . . . . . 7 | |
17 | 15, 16 | eqtr3i 2140 | . . . . . 6 |
18 | elex 2671 | . . . . . 6 | |
19 | 17, 18 | eqeltrid 2204 | . . . . 5 |
20 | 0ex 4025 | . . . . . . 7 | |
21 | dmeq 4709 | . . . . . . . . . . . . 13 | |
22 | 21 | eqeq1d 2126 | . . . . . . . . . . . 12 |
23 | fveq1 5388 | . . . . . . . . . . . . . 14 | |
24 | 23 | fveq2d 5393 | . . . . . . . . . . . . 13 |
25 | 24 | eleq2d 2187 | . . . . . . . . . . . 12 |
26 | 22, 25 | anbi12d 464 | . . . . . . . . . . 11 |
27 | 26 | rexbidv 2415 | . . . . . . . . . 10 |
28 | 21 | eqeq1d 2126 | . . . . . . . . . . 11 |
29 | 28 | anbi1d 460 | . . . . . . . . . 10 |
30 | 27, 29 | orbi12d 767 | . . . . . . . . 9 |
31 | 30 | abbidv 2235 | . . . . . . . 8 |
32 | eqid 2117 | . . . . . . . 8 | |
33 | 31, 32 | fvmptg 5465 | . . . . . . 7 |
34 | 20, 33 | mpan 420 | . . . . . 6 |
35 | 34, 17 | syl6eq 2166 | . . . . 5 |
36 | 19, 35 | syl 14 | . . . 4 |
37 | 36, 18 | eqeltrd 2194 | . . 3 |
38 | df-frec 6256 | . . . . . 6 frec recs | |
39 | 38 | fveq1i 5390 | . . . . 5 frec recs |
40 | peano1 4478 | . . . . . 6 | |
41 | fvres 5413 | . . . . . 6 recs recs | |
42 | 40, 41 | ax-mp 5 | . . . . 5 recs recs |
43 | 39, 42 | eqtri 2138 | . . . 4 frec recs |
44 | eqid 2117 | . . . . 5 recs recs | |
45 | 44 | tfr0 6188 | . . . 4 recs |
46 | 43, 45 | syl5eq 2162 | . . 3 frec |
47 | 37, 46 | syl 14 | . 2 frec |
48 | 47, 36 | eqtrd 2150 | 1 frec |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 682 wceq 1316 wcel 1465 cab 2103 wne 2285 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-in1 588 ax-in2 589 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-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-res 4521 df-iota 5058 df-fun 5095 df-fn 5096 df-fv 5101 df-recs 6170 df-frec 6256 |
This theorem is referenced by: frecrdg 6273 frec2uz0d 10140 frec2uzrdg 10150 frecuzrdg0 10154 frecuzrdgg 10157 frecuzrdg0t 10163 seq3val 10199 seqvalcd 10200 |
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