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Mirrors > Home > ILE Home > Th. List > tfrcldm | Unicode version |
Description: Recursion is defined on an ordinal if the characteristic function satisfies a closure hypothesis up to a suitable point. (Contributed by Jim Kingdon, 26-Mar-2022.) |
Ref | Expression |
---|---|
tfrcl.f | recs |
tfrcl.g | |
tfrcl.x | |
tfrcl.ex | |
tfrcl.u | |
tfrcl.yx |
Ref | Expression |
---|---|
tfrcldm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrcl.yx | . . 3 | |
2 | eluni 3734 | . . 3 | |
3 | 1, 2 | sylib 121 | . 2 |
4 | tfrcl.f | . . . 4 recs | |
5 | tfrcl.g | . . . . 5 | |
6 | 5 | adantr 274 | . . . 4 |
7 | tfrcl.x | . . . . 5 | |
8 | 7 | adantr 274 | . . . 4 |
9 | tfrcl.ex | . . . . 5 | |
10 | 9 | 3adant1r 1209 | . . . 4 |
11 | feq2 5251 | . . . . . . . 8 | |
12 | raleq 2624 | . . . . . . . 8 | |
13 | 11, 12 | anbi12d 464 | . . . . . . 7 |
14 | 13 | cbvrexv 2653 | . . . . . 6 |
15 | fveq2 5414 | . . . . . . . . . 10 | |
16 | reseq2 4809 | . . . . . . . . . . 11 | |
17 | 16 | fveq2d 5418 | . . . . . . . . . 10 |
18 | 15, 17 | eqeq12d 2152 | . . . . . . . . 9 |
19 | 18 | cbvralv 2652 | . . . . . . . 8 |
20 | 19 | anbi2i 452 | . . . . . . 7 |
21 | 20 | rexbii 2440 | . . . . . 6 |
22 | 14, 21 | bitri 183 | . . . . 5 |
23 | 22 | abbii 2253 | . . . 4 |
24 | tfrcl.u | . . . . 5 | |
25 | 24 | adantlr 468 | . . . 4 |
26 | simprr 521 | . . . 4 | |
27 | 4, 6, 8, 10, 23, 25, 26 | tfrcllemres 6252 | . . 3 |
28 | simprl 520 | . . 3 | |
29 | 27, 28 | sseldd 3093 | . 2 |
30 | 3, 29 | exlimddv 1870 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wex 1468 wcel 1480 cab 2123 wral 2414 wrex 2415 cuni 3731 word 4279 csuc 4282 cdm 4534 cres 4536 wfun 5112 wf 5114 cfv 5118 recscrecs 6194 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-recs 6195 |
This theorem is referenced by: tfrcl 6254 frecfcllem 6294 frecsuclem 6296 |
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