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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 6362. (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 4483 | . . . 4 | |
5 | 3, 4 | syl 14 | . . 3 |
6 | rdgival 6359 | . . 3 | |
7 | 1, 2, 5, 6 | syl3anc 1233 | . 2 |
8 | df-suc 4354 | . . . . . . 7 | |
9 | iuneq1 3884 | . . . . . . 7 | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 |
11 | iunxun 3950 | . . . . . 6 | |
12 | 10, 11 | eqtri 2191 | . . . . 5 |
13 | fveq2 5494 | . . . . . . . 8 | |
14 | 13 | fveq2d 5498 | . . . . . . 7 |
15 | 14 | iunxsng 3946 | . . . . . 6 |
16 | 15 | uneq2d 3281 | . . . . 5 |
17 | 12, 16 | eqtrid 2215 | . . . 4 |
18 | 17 | uneq2d 3281 | . . 3 |
19 | 3, 18 | syl 14 | . 2 |
20 | 7, 19 | eqtrd 2203 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1348 wcel 2141 cvv 2730 cun 3119 csn 3581 ciun 3871 con0 4346 csuc 4348 wfn 5191 cfv 5196 crdg 6346 |
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 4102 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 |
This theorem depends on definitions: df-bi 116 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-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 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-suc 4354 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-recs 6282 df-irdg 6347 |
This theorem is referenced by: rdgisucinc 6362 |
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