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Mirrors > Home > ILE Home > Th. List > frecuzrdgfunlem | Unicode version |
Description: The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.) |
Ref | Expression |
---|---|
frecuzrdgrclt.c | |
frecuzrdgrclt.a | |
frecuzrdgrclt.t | |
frecuzrdgrclt.f | |
frecuzrdgrclt.r | frec |
frecuzrdgfunlem.g | frec |
Ref | Expression |
---|---|
frecuzrdgfunlem |
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 10309 | . . . . 5 |
7 | frn 5327 | . . . . 5 | |
8 | 6, 7 | syl 14 | . . . 4 |
9 | xpss 4693 | . . . 4 | |
10 | 8, 9 | sstrdi 3140 | . . 3 |
11 | df-rel 4592 | . . 3 | |
12 | 10, 11 | sylibr 133 | . 2 |
13 | frecuzrdgfunlem.g | . . . . . . . . . 10 frec | |
14 | 1, 13 | frec2uzf1od 10300 | . . . . . . . . 9 |
15 | f1ocnvdm 5728 | . . . . . . . . 9 | |
16 | 14, 15 | sylan 281 | . . . . . . . 8 |
17 | 6 | ffvelrnda 5601 | . . . . . . . 8 |
18 | 16, 17 | syldan 280 | . . . . . . 7 |
19 | xp2nd 6111 | . . . . . . 7 | |
20 | 18, 19 | syl 14 | . . . . . 6 |
21 | ffn 5318 | . . . . . . . . . 10 | |
22 | fvelrnb 5515 | . . . . . . . . . 10 | |
23 | 6, 21, 22 | 3syl 17 | . . . . . . . . 9 |
24 | 6 | ffvelrnda 5601 | . . . . . . . . . . . . . . . . . . 19 |
25 | 1st2nd2 6120 | . . . . . . . . . . . . . . . . . . 19 | |
26 | 24, 25 | syl 14 | . . . . . . . . . . . . . . . . . 18 |
27 | 1 | adantr 274 | . . . . . . . . . . . . . . . . . . . 20 |
28 | 2 | adantr 274 | . . . . . . . . . . . . . . . . . . . 20 |
29 | 3 | adantr 274 | . . . . . . . . . . . . . . . . . . . 20 |
30 | 4 | adantlr 469 | . . . . . . . . . . . . . . . . . . . 20 |
31 | simpr 109 | . . . . . . . . . . . . . . . . . . . 20 | |
32 | 27, 28, 29, 30, 5, 31, 13 | frecuzrdgg 10310 | . . . . . . . . . . . . . . . . . . 19 |
33 | 32 | opeq1d 3747 | . . . . . . . . . . . . . . . . . 18 |
34 | 26, 33 | eqtrd 2190 | . . . . . . . . . . . . . . . . 17 |
35 | 34 | eqeq1d 2166 | . . . . . . . . . . . . . . . 16 |
36 | vex 2715 | . . . . . . . . . . . . . . . . . 18 | |
37 | vex 2715 | . . . . . . . . . . . . . . . . . 18 | |
38 | 36, 37 | opth2 4200 | . . . . . . . . . . . . . . . . 17 |
39 | 38 | simplbi 272 | . . . . . . . . . . . . . . . 16 |
40 | 35, 39 | syl6bi 162 | . . . . . . . . . . . . . . 15 |
41 | f1ocnvfv 5726 | . . . . . . . . . . . . . . . 16 | |
42 | 14, 41 | sylan 281 | . . . . . . . . . . . . . . 15 |
43 | 40, 42 | syld 45 | . . . . . . . . . . . . . 14 |
44 | fveq2 5467 | . . . . . . . . . . . . . . 15 | |
45 | 44 | fveq2d 5471 | . . . . . . . . . . . . . 14 |
46 | 43, 45 | syl6 33 | . . . . . . . . . . . . 13 |
47 | 46 | imp 123 | . . . . . . . . . . . 12 |
48 | 36, 37 | op2ndd 6094 | . . . . . . . . . . . . 13 |
49 | 48 | adantl 275 | . . . . . . . . . . . 12 |
50 | 47, 49 | eqtr2d 2191 | . . . . . . . . . . 11 |
51 | 50 | ex 114 | . . . . . . . . . 10 |
52 | 51 | rexlimdva 2574 | . . . . . . . . 9 |
53 | 23, 52 | sylbid 149 | . . . . . . . 8 |
54 | 53 | alrimiv 1854 | . . . . . . 7 |
55 | 54 | adantr 274 | . . . . . 6 |
56 | eqeq2 2167 | . . . . . . . . 9 | |
57 | 56 | imbi2d 229 | . . . . . . . 8 |
58 | 57 | albidv 1804 | . . . . . . 7 |
59 | 58 | spcegv 2800 | . . . . . 6 |
60 | 20, 55, 59 | sylc 62 | . . . . 5 |
61 | nfv 1508 | . . . . . 6 | |
62 | 61 | mo2r 2058 | . . . . 5 |
63 | 60, 62 | syl 14 | . . . 4 |
64 | 1, 2, 3, 4, 5 | frecuzrdgdom 10312 | . . . . . 6 |
65 | 64 | eleq2d 2227 | . . . . 5 |
66 | 65 | pm5.32i 450 | . . . 4 |
67 | df-br 3966 | . . . . 5 | |
68 | 67 | mobii 2043 | . . . 4 |
69 | 63, 66, 68 | 3imtr4i 200 | . . 3 |
70 | 69 | ralrimiva 2530 | . 2 |
71 | dffun7 5196 | . 2 | |
72 | 12, 70, 71 | sylanbrc 414 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1333 wceq 1335 wex 1472 wmo 2007 wcel 2128 wral 2435 wrex 2436 cvv 2712 wss 3102 cop 3563 class class class wbr 3965 cmpt 4025 com 4548 cxp 4583 ccnv 4584 cdm 4585 crn 4586 wrel 4590 wfun 5163 wfn 5164 wf 5165 wf1o 5168 cfv 5169 (class class class)co 5821 cmpo 5823 c1st 6083 c2nd 6084 freccfrec 6334 c1 7728 caddc 7730 cz 9162 cuz 9434 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546 ax-cnex 7818 ax-resscn 7819 ax-1cn 7820 ax-1re 7821 ax-icn 7822 ax-addcl 7823 ax-addrcl 7824 ax-mulcl 7825 ax-addcom 7827 ax-addass 7829 ax-distr 7831 ax-i2m1 7832 ax-0lt1 7833 ax-0id 7835 ax-rnegex 7836 ax-cnre 7838 ax-pre-ltirr 7839 ax-pre-ltwlin 7840 ax-pre-lttrn 7841 ax-pre-ltadd 7843 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-recs 6249 df-frec 6335 df-pnf 7909 df-mnf 7910 df-xr 7911 df-ltxr 7912 df-le 7913 df-sub 8043 df-neg 8044 df-inn 8829 df-n0 9086 df-z 9163 df-uz 9435 |
This theorem is referenced by: frecuzrdgfun 10314 |
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