Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > frecuzrdgdomlem | Unicode version |
Description: The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.) |
Ref | Expression |
---|---|
frecuzrdgrclt.c | |
frecuzrdgrclt.a | |
frecuzrdgrclt.t | |
frecuzrdgrclt.f | |
frecuzrdgrclt.r | frec |
frecuzrdgdomlem.g | frec |
Ref | Expression |
---|---|
frecuzrdgdomlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frecuzrdgrclt.c | . . . . . 6 | |
2 | frecuzrdgrclt.a | . . . . . 6 | |
3 | frecuzrdgrclt.t | . . . . . 6 | |
4 | frecuzrdgrclt.f | . . . . . 6 | |
5 | frecuzrdgrclt.r | . . . . . 6 frec | |
6 | 1, 2, 3, 4, 5 | frecuzrdgrclt 10156 | . . . . 5 |
7 | frn 5251 | . . . . 5 | |
8 | 6, 7 | syl 14 | . . . 4 |
9 | dmss 4708 | . . . 4 | |
10 | 8, 9 | syl 14 | . . 3 |
11 | dmxpss 4939 | . . 3 | |
12 | 10, 11 | sstrdi 3079 | . 2 |
13 | 8 | adantr 274 | . . . . . . . . 9 |
14 | ffun 5245 | . . . . . . . . . . . 12 | |
15 | 6, 14 | syl 14 | . . . . . . . . . . 11 |
16 | 15 | adantr 274 | . . . . . . . . . 10 |
17 | frecuzrdgdomlem.g | . . . . . . . . . . . . 13 frec | |
18 | 1, 17 | frec2uzf1od 10147 | . . . . . . . . . . . 12 |
19 | f1ocnvdm 5650 | . . . . . . . . . . . 12 | |
20 | 18, 19 | sylan 281 | . . . . . . . . . . 11 |
21 | fdm 5248 | . . . . . . . . . . . . 13 | |
22 | 6, 21 | syl 14 | . . . . . . . . . . . 12 |
23 | 22 | adantr 274 | . . . . . . . . . . 11 |
24 | 20, 23 | eleqtrrd 2197 | . . . . . . . . . 10 |
25 | fvelrn 5519 | . . . . . . . . . 10 | |
26 | 16, 24, 25 | syl2anc 408 | . . . . . . . . 9 |
27 | 13, 26 | sseldd 3068 | . . . . . . . 8 |
28 | 1st2nd2 6041 | . . . . . . . 8 | |
29 | 27, 28 | syl 14 | . . . . . . 7 |
30 | 1 | adantr 274 | . . . . . . . . . 10 |
31 | 2 | adantr 274 | . . . . . . . . . 10 |
32 | 3 | adantr 274 | . . . . . . . . . 10 |
33 | 4 | adantlr 468 | . . . . . . . . . 10 |
34 | 30, 31, 32, 33, 5, 20, 17 | frecuzrdgg 10157 | . . . . . . . . 9 |
35 | f1ocnvfv2 5647 | . . . . . . . . . 10 | |
36 | 18, 35 | sylan 281 | . . . . . . . . 9 |
37 | 34, 36 | eqtrd 2150 | . . . . . . . 8 |
38 | 37 | opeq1d 3681 | . . . . . . 7 |
39 | 29, 38 | eqtrd 2150 | . . . . . 6 |
40 | 39, 26 | eqeltrrd 2195 | . . . . 5 |
41 | simpr 109 | . . . . . 6 | |
42 | xp2nd 6032 | . . . . . . 7 | |
43 | 27, 42 | syl 14 | . . . . . 6 |
44 | opeldmg 4714 | . . . . . 6 | |
45 | 41, 43, 44 | syl2anc 408 | . . . . 5 |
46 | 40, 45 | mpd 13 | . . . 4 |
47 | 46 | ex 114 | . . 3 |
48 | 47 | ssrdv 3073 | . 2 |
49 | 12, 48 | eqssd 3084 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 wss 3041 cop 3500 cmpt 3959 com 4474 cxp 4507 ccnv 4508 cdm 4509 crn 4510 wfun 5087 wf 5089 wf1o 5092 cfv 5093 (class class class)co 5742 cmpo 5744 c1st 6004 c2nd 6005 freccfrec 6255 c1 7589 caddc 7591 cz 9022 cuz 9294 |
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-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 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-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 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-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 df-uz 9295 |
This theorem is referenced by: frecuzrdgdom 10159 |
Copyright terms: Public domain | W3C validator |