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Mirrors > Home > ILE Home > Th. List > tfri2d | Unicode version |
Description: Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule ( as described at tfri1 5974). Here we show that the function has the property that for any function satisfying that condition, the "next" value of is recursively applied to all "previous" values of . (Contributed by Jim Kingdon, 4-May-2019.) |
Ref | Expression |
---|---|
tfri1d.1 | recs |
tfri1d.2 |
Ref | Expression |
---|---|
tfri2d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfri1d.1 | . . . . . 6 recs | |
2 | tfri1d.2 | . . . . . 6 | |
3 | 1, 2 | tfri1d 5972 | . . . . 5 |
4 | fndm 5018 | . . . . 5 | |
5 | 3, 4 | syl 14 | . . . 4 |
6 | 5 | eleq2d 2148 | . . 3 |
7 | 6 | biimpar 291 | . 2 |
8 | 1 | tfr2a 5959 | . 2 |
9 | 7, 8 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wal 1282 wceq 1284 wcel 1433 cvv 2601 con0 4118 cdm 4363 cres 4365 wfun 4916 wfn 4917 cfv 4922 recscrecs 5942 |
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 |
This theorem is referenced by: rdgivallem 5991 frecsuclem1 6010 |
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