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Mirrors > Home > ILE Home > Th. List > rdgisuc1 | Unicode version |
Description: One way of describing the
value of the recursive definition generator at
a successor. There is no condition on the characteristic function
other than
. Given that, the resulting expression
encompasses both the expected successor term
but also
terms that correspond to
the initial value and to limit ordinals
.
If we add conditions on the characteristic function, we can show tighter results such as rdgisucinc 5995. (Contributed by Jim Kingdon, 9-Jun-2019.) |
Ref | Expression |
---|---|
rdgisuc1.1 | |
rdgisuc1.2 | |
rdgisuc1.3 |
Ref | Expression |
---|---|
rdgisuc1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgisuc1.1 | . . 3 | |
2 | rdgisuc1.2 | . . 3 | |
3 | rdgisuc1.3 | . . . 4 | |
4 | suceloni 4245 | . . . 4 | |
5 | 3, 4 | syl 14 | . . 3 |
6 | rdgival 5992 | . . 3 | |
7 | 1, 2, 5, 6 | syl3anc 1169 | . 2 |
8 | df-suc 4126 | . . . . . . 7 | |
9 | iuneq1 3691 | . . . . . . 7 | |
10 | 8, 9 | ax-mp 7 | . . . . . 6 |
11 | iunxun 3756 | . . . . . 6 | |
12 | 10, 11 | eqtri 2101 | . . . . 5 |
13 | fveq2 5198 | . . . . . . . 8 | |
14 | 13 | fveq2d 5202 | . . . . . . 7 |
15 | 14 | iunxsng 3753 | . . . . . 6 |
16 | 15 | uneq2d 3126 | . . . . 5 |
17 | 12, 16 | syl5eq 2125 | . . . 4 |
18 | 17 | uneq2d 3126 | . . 3 |
19 | 3, 18 | syl 14 | . 2 |
20 | 7, 19 | eqtrd 2113 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1284 wcel 1433 cvv 2601 cun 2971 csn 3398 ciun 3678 con0 4118 csuc 4120 wfn 4917 cfv 4922 crdg 5979 |
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-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 |
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-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-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 |
This theorem is referenced by: rdgisucinc 5995 |
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