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

Theorem rdgfun 5881
Description: The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
rdgfun Fun rec(𝐹, A)

Proof of Theorem rdgfun
Dummy variables x y z f g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2022 . . 3 {fy On (f Fn y z y (fz) = ((g V ↦ (A x dom g(𝐹‘(gx))))‘(fz)))} = {fy On (f Fn y z y (fz) = ((g V ↦ (A x dom g(𝐹‘(gx))))‘(fz)))}
21tfrlem7 5855 . 2 Fun recs((g V ↦ (A x dom g(𝐹‘(gx)))))
3 df-irdg 5878 . . 3 rec(𝐹, A) = recs((g V ↦ (A x dom g(𝐹‘(gx)))))
43funeqi 4848 . 2 (Fun rec(𝐹, A) ↔ Fun recs((g V ↦ (A x dom g(𝐹‘(gx))))))
52, 4mpbir 134 1 Fun rec(𝐹, A)
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1228  {cab 2008  wral 2284  wrex 2285  Vcvv 2535  cun 2892   ciun 3631  cmpt 3792  Oncon0 4049  dom cdm 4272  cres 4274  Fun wfun 4823   Fn wfn 4824  cfv 4829  recscrecs 5841  reccrdg 5877
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-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-setind 4204
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-res 4284  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837  df-recs 5842  df-irdg 5878
This theorem is referenced by:  rdgivallem  5888
  Copyright terms: Public domain W3C validator