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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 10371 | . . . . 5 |
7 | frn 5356 | . . . . 5 | |
8 | 6, 7 | syl 14 | . . . 4 |
9 | xpss 4719 | . . . 4 | |
10 | 8, 9 | sstrdi 3159 | . . 3 |
11 | df-rel 4618 | . . 3 | |
12 | 10, 11 | sylibr 133 | . 2 |
13 | frecuzrdgfunlem.g | . . . . . . . . . 10 frec | |
14 | 1, 13 | frec2uzf1od 10362 | . . . . . . . . 9 |
15 | f1ocnvdm 5760 | . . . . . . . . 9 | |
16 | 14, 15 | sylan 281 | . . . . . . . 8 |
17 | 6 | ffvelrnda 5631 | . . . . . . . 8 |
18 | 16, 17 | syldan 280 | . . . . . . 7 |
19 | xp2nd 6145 | . . . . . . 7 | |
20 | 18, 19 | syl 14 | . . . . . 6 |
21 | ffn 5347 | . . . . . . . . . 10 | |
22 | fvelrnb 5544 | . . . . . . . . . 10 | |
23 | 6, 21, 22 | 3syl 17 | . . . . . . . . 9 |
24 | 6 | ffvelrnda 5631 | . . . . . . . . . . . . . . . . . . 19 |
25 | 1st2nd2 6154 | . . . . . . . . . . . . . . . . . . 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 474 | . . . . . . . . . . . . . . . . . . . 20 |
31 | simpr 109 | . . . . . . . . . . . . . . . . . . . 20 | |
32 | 27, 28, 29, 30, 5, 31, 13 | frecuzrdgg 10372 | . . . . . . . . . . . . . . . . . . 19 |
33 | 32 | opeq1d 3771 | . . . . . . . . . . . . . . . . . 18 |
34 | 26, 33 | eqtrd 2203 | . . . . . . . . . . . . . . . . 17 |
35 | 34 | eqeq1d 2179 | . . . . . . . . . . . . . . . 16 |
36 | vex 2733 | . . . . . . . . . . . . . . . . . 18 | |
37 | vex 2733 | . . . . . . . . . . . . . . . . . 18 | |
38 | 36, 37 | opth2 4225 | . . . . . . . . . . . . . . . . 17 |
39 | 38 | simplbi 272 | . . . . . . . . . . . . . . . 16 |
40 | 35, 39 | syl6bi 162 | . . . . . . . . . . . . . . 15 |
41 | f1ocnvfv 5758 | . . . . . . . . . . . . . . . 16 | |
42 | 14, 41 | sylan 281 | . . . . . . . . . . . . . . 15 |
43 | 40, 42 | syld 45 | . . . . . . . . . . . . . 14 |
44 | fveq2 5496 | . . . . . . . . . . . . . . 15 | |
45 | 44 | fveq2d 5500 | . . . . . . . . . . . . . 14 |
46 | 43, 45 | syl6 33 | . . . . . . . . . . . . 13 |
47 | 46 | imp 123 | . . . . . . . . . . . 12 |
48 | 36, 37 | op2ndd 6128 | . . . . . . . . . . . . 13 |
49 | 48 | adantl 275 | . . . . . . . . . . . 12 |
50 | 47, 49 | eqtr2d 2204 | . . . . . . . . . . 11 |
51 | 50 | ex 114 | . . . . . . . . . 10 |
52 | 51 | rexlimdva 2587 | . . . . . . . . 9 |
53 | 23, 52 | sylbid 149 | . . . . . . . 8 |
54 | 53 | alrimiv 1867 | . . . . . . 7 |
55 | 54 | adantr 274 | . . . . . 6 |
56 | eqeq2 2180 | . . . . . . . . 9 | |
57 | 56 | imbi2d 229 | . . . . . . . 8 |
58 | 57 | albidv 1817 | . . . . . . 7 |
59 | 58 | spcegv 2818 | . . . . . 6 |
60 | 20, 55, 59 | sylc 62 | . . . . 5 |
61 | nfv 1521 | . . . . . 6 | |
62 | 61 | mo2r 2071 | . . . . 5 |
63 | 60, 62 | syl 14 | . . . 4 |
64 | 1, 2, 3, 4, 5 | frecuzrdgdom 10374 | . . . . . 6 |
65 | 64 | eleq2d 2240 | . . . . 5 |
66 | 65 | pm5.32i 451 | . . . 4 |
67 | df-br 3990 | . . . . 5 | |
68 | 67 | mobii 2056 | . . . 4 |
69 | 63, 66, 68 | 3imtr4i 200 | . . 3 |
70 | 69 | ralrimiva 2543 | . 2 |
71 | dffun7 5225 | . 2 | |
72 | 12, 70, 71 | sylanbrc 415 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1346 wceq 1348 wex 1485 wmo 2020 wcel 2141 wral 2448 wrex 2449 cvv 2730 wss 3121 cop 3586 class class class wbr 3989 cmpt 4050 com 4574 cxp 4609 ccnv 4610 cdm 4611 crn 4612 wrel 4616 wfun 5192 wfn 5193 wf 5194 wf1o 5197 cfv 5198 (class class class)co 5853 cmpo 5855 c1st 6117 c2nd 6118 freccfrec 6369 c1 7775 caddc 7777 cz 9212 cuz 9487 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 |
This theorem is referenced by: frecuzrdgfun 10376 |
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