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Mirrors > Home > ILE Home > Th. List > frecrdg | Unicode version |
Description: Transfinite recursion
restricted to omega.
Given a suitable characteristic function, df-frec 6001 produces the same results as df-irdg 5980 restricted to . Presumably the theorem would also hold if were changed to . (Contributed by Jim Kingdon, 29-Aug-2019.) |
Ref | Expression |
---|---|
frecrdg.1 | |
frecrdg.2 | |
frecrdg.inc |
Ref | Expression |
---|---|
frecrdg | frec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frecrdg.1 | . . . 4 | |
2 | vex 2604 | . . . . . 6 | |
3 | funfvex 5212 | . . . . . . 7 | |
4 | 3 | funfni 5019 | . . . . . 6 |
5 | 2, 4 | mpan2 415 | . . . . 5 |
6 | 5 | alrimiv 1795 | . . . 4 |
7 | 1, 6 | syl 14 | . . 3 |
8 | frecrdg.2 | . . 3 | |
9 | frecfnom 6009 | . . 3 frec | |
10 | 7, 8, 9 | syl2anc 403 | . 2 frec |
11 | rdgifnon2 5990 | . . . 4 | |
12 | 7, 8, 11 | syl2anc 403 | . . 3 |
13 | omsson 4353 | . . 3 | |
14 | fnssres 5032 | . . 3 | |
15 | 12, 13, 14 | sylancl 404 | . 2 |
16 | fveq2 5198 | . . . . 5 frec frec | |
17 | fveq2 5198 | . . . . 5 | |
18 | 16, 17 | eqeq12d 2095 | . . . 4 frec frec |
19 | fveq2 5198 | . . . . 5 frec frec | |
20 | fveq2 5198 | . . . . 5 | |
21 | 19, 20 | eqeq12d 2095 | . . . 4 frec frec |
22 | fveq2 5198 | . . . . 5 frec frec | |
23 | fveq2 5198 | . . . . 5 | |
24 | 22, 23 | eqeq12d 2095 | . . . 4 frec frec |
25 | frec0g 6006 | . . . . . 6 frec | |
26 | 8, 25 | syl 14 | . . . . 5 frec |
27 | peano1 4335 | . . . . . . 7 | |
28 | fvres 5219 | . . . . . . 7 | |
29 | 27, 28 | ax-mp 7 | . . . . . 6 |
30 | rdg0g 5998 | . . . . . . 7 | |
31 | 8, 30 | syl 14 | . . . . . 6 |
32 | 29, 31 | syl5eq 2125 | . . . . 5 |
33 | 26, 32 | eqtr4d 2116 | . . . 4 frec |
34 | simpr 108 | . . . . . . . . . 10 frec frec | |
35 | fvres 5219 | . . . . . . . . . . 11 | |
36 | 35 | ad2antlr 472 | . . . . . . . . . 10 frec |
37 | 34, 36 | eqtrd 2113 | . . . . . . . . 9 frec frec |
38 | 37 | fveq2d 5202 | . . . . . . . 8 frec frec |
39 | 7, 8 | jca 300 | . . . . . . . . . 10 |
40 | frecsuc 6014 | . . . . . . . . . . 11 frec frec | |
41 | 40 | 3expa 1138 | . . . . . . . . . 10 frec frec |
42 | 39, 41 | sylan 277 | . . . . . . . . 9 frec frec |
43 | 42 | adantr 270 | . . . . . . . 8 frec frec frec |
44 | 1 | adantr 270 | . . . . . . . . . 10 |
45 | 8 | adantr 270 | . . . . . . . . . 10 |
46 | simpr 108 | . . . . . . . . . . 11 | |
47 | nnon 4350 | . . . . . . . . . . 11 | |
48 | 46, 47 | syl 14 | . . . . . . . . . 10 |
49 | frecrdg.inc | . . . . . . . . . . 11 | |
50 | 49 | adantr 270 | . . . . . . . . . 10 |
51 | 44, 45, 48, 50 | rdgisucinc 5995 | . . . . . . . . 9 |
52 | 51 | adantr 270 | . . . . . . . 8 frec |
53 | 38, 43, 52 | 3eqtr4d 2123 | . . . . . . 7 frec frec |
54 | peano2 4336 | . . . . . . . . 9 | |
55 | fvres 5219 | . . . . . . . . 9 | |
56 | 54, 55 | syl 14 | . . . . . . . 8 |
57 | 56 | ad2antlr 472 | . . . . . . 7 frec |
58 | 53, 57 | eqtr4d 2116 | . . . . . 6 frec frec |
59 | 58 | ex 113 | . . . . 5 frec frec |
60 | 59 | expcom 114 | . . . 4 frec frec |
61 | 18, 21, 24, 33, 60 | finds2 4342 | . . 3 frec |
62 | 61 | impcom 123 | . 2 frec |
63 | 10, 15, 62 | eqfnfvd 5289 | 1 frec |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wal 1282 wceq 1284 wcel 1433 cvv 2601 wss 2973 c0 3251 con0 4118 csuc 4120 com 4331 cres 4365 wfn 4917 cfv 4922 crdg 5979 freccfrec 6000 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-recs 5943 df-irdg 5980 df-frec 6001 |
This theorem is referenced by: (None) |
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