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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 5978 produces the same results as df-irdg 5957 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 2560 | . . . . . 6 | |
3 | funfvex 5192 | . . . . . . 7 | |
4 | 3 | funfni 4999 | . . . . . 6 |
5 | 2, 4 | mpan2 401 | . . . . 5 |
6 | 5 | alrimiv 1754 | . . . 4 |
7 | 1, 6 | syl 14 | . . 3 |
8 | frecrdg.2 | . . 3 | |
9 | frecfnom 5986 | . . 3 frec | |
10 | 7, 8, 9 | syl2anc 391 | . 2 frec |
11 | rdgifnon2 5967 | . . . 4 | |
12 | 7, 8, 11 | syl2anc 391 | . . 3 |
13 | omsson 4335 | . . 3 | |
14 | fnssres 5012 | . . 3 | |
15 | 12, 13, 14 | sylancl 392 | . 2 |
16 | fveq2 5178 | . . . . 5 frec frec | |
17 | fveq2 5178 | . . . . 5 | |
18 | 16, 17 | eqeq12d 2054 | . . . 4 frec frec |
19 | fveq2 5178 | . . . . 5 frec frec | |
20 | fveq2 5178 | . . . . 5 | |
21 | 19, 20 | eqeq12d 2054 | . . . 4 frec frec |
22 | fveq2 5178 | . . . . 5 frec frec | |
23 | fveq2 5178 | . . . . 5 | |
24 | 22, 23 | eqeq12d 2054 | . . . 4 frec frec |
25 | frec0g 5983 | . . . . . 6 frec | |
26 | 8, 25 | syl 14 | . . . . 5 frec |
27 | peano1 4317 | . . . . . . 7 | |
28 | fvres 5198 | . . . . . . 7 | |
29 | 27, 28 | ax-mp 7 | . . . . . 6 |
30 | rdg0g 5975 | . . . . . . 7 | |
31 | 8, 30 | syl 14 | . . . . . 6 |
32 | 29, 31 | syl5eq 2084 | . . . . 5 |
33 | 26, 32 | eqtr4d 2075 | . . . 4 frec |
34 | simpr 103 | . . . . . . . . . 10 frec frec | |
35 | fvres 5198 | . . . . . . . . . . 11 | |
36 | 35 | ad2antlr 458 | . . . . . . . . . 10 frec |
37 | 34, 36 | eqtrd 2072 | . . . . . . . . 9 frec frec |
38 | 37 | fveq2d 5182 | . . . . . . . 8 frec frec |
39 | 7, 8 | jca 290 | . . . . . . . . . 10 |
40 | frecsuc 5991 | . . . . . . . . . . 11 frec frec | |
41 | 40 | 3expa 1104 | . . . . . . . . . 10 frec frec |
42 | 39, 41 | sylan 267 | . . . . . . . . 9 frec frec |
43 | 42 | adantr 261 | . . . . . . . 8 frec frec frec |
44 | 1 | adantr 261 | . . . . . . . . . 10 |
45 | 8 | adantr 261 | . . . . . . . . . 10 |
46 | simpr 103 | . . . . . . . . . . 11 | |
47 | nnon 4332 | . . . . . . . . . . 11 | |
48 | 46, 47 | syl 14 | . . . . . . . . . 10 |
49 | frecrdg.inc | . . . . . . . . . . 11 | |
50 | 49 | adantr 261 | . . . . . . . . . 10 |
51 | 44, 45, 48, 50 | rdgisucinc 5972 | . . . . . . . . 9 |
52 | 51 | adantr 261 | . . . . . . . 8 frec |
53 | 38, 43, 52 | 3eqtr4d 2082 | . . . . . . 7 frec frec |
54 | peano2 4318 | . . . . . . . . 9 | |
55 | fvres 5198 | . . . . . . . . 9 | |
56 | 54, 55 | syl 14 | . . . . . . . 8 |
57 | 56 | ad2antlr 458 | . . . . . . 7 frec |
58 | 53, 57 | eqtr4d 2075 | . . . . . 6 frec frec |
59 | 58 | ex 108 | . . . . 5 frec frec |
60 | 59 | expcom 109 | . . . 4 frec frec |
61 | 18, 21, 24, 33, 60 | finds2 4324 | . . 3 frec |
62 | 61 | impcom 116 | . 2 frec |
63 | 10, 15, 62 | eqfnfvd 5268 | 1 frec |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wal 1241 wceq 1243 wcel 1393 cvv 2557 wss 2917 c0 3224 con0 4100 csuc 4102 com 4313 cres 4347 wfn 4897 cfv 4902 crdg 5956 freccfrec 5977 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-recs 5920 df-irdg 5957 df-frec 5978 |
This theorem is referenced by: (None) |
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