ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rdgifnon Structured version   GIF version

Theorem rdgifnon 5903
Description: The recursive definition generator is a function on ordinal numbers. The 𝐹 Fn V condition states that the characteristic function is defined for all sets (being defined for all ordinals might be enough if being used in a manner similar to rdgon 5910; in cases like df-oadd 5937 either presumably could work). (Contributed by Jim Kingdon, 13-Jul-2019.)
Assertion
Ref Expression
rdgifnon ((𝐹 Fn V A 𝑉) → rec(𝐹, A) Fn On)

Proof of Theorem rdgifnon
Dummy variables f g x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-irdg 5894 . 2 rec(𝐹, A) = recs((g V ↦ (A x dom g(𝐹‘(gx)))))
2 rdgruledefgg 5899 . . 3 ((𝐹 Fn V A 𝑉) → (Fun (g V ↦ (A x dom g(𝐹‘(gx)))) ((g V ↦ (A x dom g(𝐹‘(gx))))‘f) V))
32alrimiv 1751 . 2 ((𝐹 Fn V A 𝑉) → f(Fun (g V ↦ (A x dom g(𝐹‘(gx)))) ((g V ↦ (A x dom g(𝐹‘(gx))))‘f) V))
41, 3tfri1d 5887 1 ((𝐹 Fn V A 𝑉) → rec(𝐹, A) Fn On)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  Vcvv 2551  cun 2909   ciun 3647  cmpt 3808  Oncon0 4065  dom cdm 4287  Fun wfun 4838   Fn wfn 4839  cfv 4844  reccrdg 5893
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3862  ax-sep 3865  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-setind 4219
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-tr 3845  df-id 4020  df-iord 4068  df-on 4070  df-suc 4073  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852  df-recs 5858  df-irdg 5894
This theorem is referenced by:  rdgivallem  5905
  Copyright terms: Public domain W3C validator