Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  rdgisuc1 Unicode version

Theorem rdgisuc1 5971
 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 5972. (Contributed by Jim Kingdon, 9-Jun-2019.)
Hypotheses
Ref Expression
rdgisuc1.1
rdgisuc1.2
rdgisuc1.3
Assertion
Ref Expression
rdgisuc1
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem rdgisuc1
StepHypRef Expression
1 rdgisuc1.1 . . 3
2 rdgisuc1.2 . . 3
3 rdgisuc1.3 . . . 4
4 suceloni 4227 . . . 4
53, 4syl 14 . . 3
6 rdgival 5969 . . 3
71, 2, 5, 6syl3anc 1135 . 2
8 df-suc 4108 . . . . . . 7
9 iuneq1 3670 . . . . . . 7
108, 9ax-mp 7 . . . . . 6
11 iunxun 3735 . . . . . 6
1210, 11eqtri 2060 . . . . 5
13 fveq2 5178 . . . . . . . 8
1413fveq2d 5182 . . . . . . 7
1514iunxsng 3732 . . . . . 6
1615uneq2d 3097 . . . . 5
1712, 16syl5eq 2084 . . . 4
1817uneq2d 3097 . . 3
193, 18syl 14 . 2
207, 19eqtrd 2072 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1243   wcel 1393  cvv 2557   cun 2915  csn 3375  ciun 3657  con0 4100   csuc 4102   wfn 4897  cfv 4902  crdg 5956 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920  df-irdg 5957 This theorem is referenced by:  rdgisucinc  5972
 Copyright terms: Public domain W3C validator