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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 10143 | . . . . 5 |
7 | frn 5251 | . . . . 5 | |
8 | 6, 7 | syl 14 | . . . 4 |
9 | xpss 4617 | . . . 4 | |
10 | 8, 9 | sstrdi 3079 | . . 3 |
11 | df-rel 4516 | . . 3 | |
12 | 10, 11 | sylibr 133 | . 2 |
13 | frecuzrdgfunlem.g | . . . . . . . . . 10 frec | |
14 | 1, 13 | frec2uzf1od 10134 | . . . . . . . . 9 |
15 | f1ocnvdm 5650 | . . . . . . . . 9 | |
16 | 14, 15 | sylan 281 | . . . . . . . 8 |
17 | 6 | ffvelrnda 5523 | . . . . . . . 8 |
18 | 16, 17 | syldan 280 | . . . . . . 7 |
19 | xp2nd 6032 | . . . . . . 7 | |
20 | 18, 19 | syl 14 | . . . . . 6 |
21 | ffn 5242 | . . . . . . . . . 10 | |
22 | fvelrnb 5437 | . . . . . . . . . 10 | |
23 | 6, 21, 22 | 3syl 17 | . . . . . . . . 9 |
24 | 6 | ffvelrnda 5523 | . . . . . . . . . . . . . . . . . . 19 |
25 | 1st2nd2 6041 | . . . . . . . . . . . . . . . . . . 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 468 | . . . . . . . . . . . . . . . . . . . 20 |
31 | simpr 109 | . . . . . . . . . . . . . . . . . . . 20 | |
32 | 27, 28, 29, 30, 5, 31, 13 | frecuzrdgg 10144 | . . . . . . . . . . . . . . . . . . 19 |
33 | 32 | opeq1d 3681 | . . . . . . . . . . . . . . . . . 18 |
34 | 26, 33 | eqtrd 2150 | . . . . . . . . . . . . . . . . 17 |
35 | 34 | eqeq1d 2126 | . . . . . . . . . . . . . . . 16 |
36 | vex 2663 | . . . . . . . . . . . . . . . . . 18 | |
37 | vex 2663 | . . . . . . . . . . . . . . . . . 18 | |
38 | 36, 37 | opth2 4132 | . . . . . . . . . . . . . . . . 17 |
39 | 38 | simplbi 272 | . . . . . . . . . . . . . . . 16 |
40 | 35, 39 | syl6bi 162 | . . . . . . . . . . . . . . 15 |
41 | f1ocnvfv 5648 | . . . . . . . . . . . . . . . 16 | |
42 | 14, 41 | sylan 281 | . . . . . . . . . . . . . . 15 |
43 | 40, 42 | syld 45 | . . . . . . . . . . . . . 14 |
44 | fveq2 5389 | . . . . . . . . . . . . . . 15 | |
45 | 44 | fveq2d 5393 | . . . . . . . . . . . . . 14 |
46 | 43, 45 | syl6 33 | . . . . . . . . . . . . 13 |
47 | 46 | imp 123 | . . . . . . . . . . . 12 |
48 | 36, 37 | op2ndd 6015 | . . . . . . . . . . . . 13 |
49 | 48 | adantl 275 | . . . . . . . . . . . 12 |
50 | 47, 49 | eqtr2d 2151 | . . . . . . . . . . 11 |
51 | 50 | ex 114 | . . . . . . . . . 10 |
52 | 51 | rexlimdva 2526 | . . . . . . . . 9 |
53 | 23, 52 | sylbid 149 | . . . . . . . 8 |
54 | 53 | alrimiv 1830 | . . . . . . 7 |
55 | 54 | adantr 274 | . . . . . 6 |
56 | eqeq2 2127 | . . . . . . . . 9 | |
57 | 56 | imbi2d 229 | . . . . . . . 8 |
58 | 57 | albidv 1780 | . . . . . . 7 |
59 | 58 | spcegv 2748 | . . . . . 6 |
60 | 20, 55, 59 | sylc 62 | . . . . 5 |
61 | nfv 1493 | . . . . . 6 | |
62 | 61 | mo2r 2029 | . . . . 5 |
63 | 60, 62 | syl 14 | . . . 4 |
64 | 1, 2, 3, 4, 5 | frecuzrdgdom 10146 | . . . . . 6 |
65 | 64 | eleq2d 2187 | . . . . 5 |
66 | 65 | pm5.32i 449 | . . . 4 |
67 | df-br 3900 | . . . . 5 | |
68 | 67 | mobii 2014 | . . . 4 |
69 | 63, 66, 68 | 3imtr4i 200 | . . 3 |
70 | 69 | ralrimiva 2482 | . 2 |
71 | dffun7 5120 | . 2 | |
72 | 12, 70, 71 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1314 wceq 1316 wex 1453 wcel 1465 wmo 1978 wral 2393 wrex 2394 cvv 2660 wss 3041 cop 3500 class class class wbr 3899 cmpt 3959 com 4474 cxp 4507 ccnv 4508 cdm 4509 crn 4510 wrel 4514 wfun 5087 wfn 5088 wf 5089 wf1o 5092 cfv 5093 (class class class)co 5742 cmpo 5744 c1st 6004 c2nd 6005 freccfrec 6255 c1 7589 caddc 7591 cz 9012 cuz 9282 |
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 8685 df-n0 8936 df-z 9013 df-uz 9283 |
This theorem is referenced by: frecuzrdgfun 10148 |
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