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Mirrors > Home > ILE Home > Th. List > frecfun | Unicode version |
Description: Finite recursion produces a function. See also frecfnom 6266 which also states that the domain of that function is but which puts conditions on and . (Contributed by Jim Kingdon, 13-Feb-2022.) |
Ref | Expression |
---|---|
frecfun | frec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrfun 6185 | . . 3 recs | |
2 | funres 5134 | . . 3 recs recs | |
3 | 1, 2 | ax-mp 5 | . 2 recs |
4 | df-frec 6256 | . . 3 frec recs | |
5 | 4 | funeqi 5114 | . 2 frec recs |
6 | 3, 5 | mpbir 145 | 1 frec |
Colors of variables: wff set class |
Syntax hints: wa 103 wo 682 wceq 1316 wcel 1465 cab 2103 wrex 2394 cvv 2660 c0 3333 cmpt 3959 csuc 4257 com 4474 cdm 4509 cres 4511 wfun 5087 cfv 5093 recscrecs 6169 freccfrec 6255 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-res 4521 df-iota 5058 df-fun 5095 df-fn 5096 df-fv 5101 df-recs 6170 df-frec 6256 |
This theorem is referenced by: (None) |
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