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Mirrors > Home > ILE Home > Th. List > tfri1d | Unicode version |
Description: Principle of Transfinite
Recursion, part 1 of 3. Theorem 7.41(1) of
[TakeutiZaring] p. 47, with an
additional condition.
The condition is that is defined "everywhere", which is stated here as . Alternately, would suffice. Given a function satisfying that condition, we define a class of all "acceptable" functions. The final function we're interested in is the union recs of them. is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of . In this first part we show that is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.) |
Ref | Expression |
---|---|
tfri1d.1 | recs |
tfri1d.2 |
Ref | Expression |
---|---|
tfri1d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2170 | . . . . . 6 | |
2 | 1 | tfrlem3 6288 | . . . . 5 |
3 | tfri1d.2 | . . . . 5 | |
4 | 2, 3 | tfrlemi14d 6310 | . . . 4 recs |
5 | eqid 2170 | . . . . 5 | |
6 | 5 | tfrlem7 6294 | . . . 4 recs |
7 | 4, 6 | jctil 310 | . . 3 recs recs |
8 | df-fn 5199 | . . 3 recs recs recs | |
9 | 7, 8 | sylibr 133 | . 2 recs |
10 | tfri1d.1 | . . 3 recs | |
11 | 10 | fneq1i 5290 | . 2 recs |
12 | 9, 11 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1346 wceq 1348 wcel 2141 cab 2156 wral 2448 wrex 2449 cvv 2730 con0 4346 cdm 4609 cres 4611 wfun 5190 wfn 5191 cfv 5196 recscrecs 6281 |
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 |
This theorem is referenced by: tfri2d 6313 tfri1 6342 rdgifnon 6356 rdgifnon2 6357 frecfnom 6378 |
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