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Theorem rdgfun 5900
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 2037 . . 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 5874 . 2 Fun recs((g V ↦ (A x dom g(𝐹‘(gx)))))
3 df-irdg 5897 . . 3 rec(𝐹, A) = recs((g V ↦ (A x dom g(𝐹‘(gx)))))
43funeqi 4865 . 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 1242  {cab 2023  wral 2300  wrex 2301  Vcvv 2551  cun 2909   ciun 3648  cmpt 3809  Oncon0 4066  dom cdm 4288  cres 4290  Fun wfun 4839   Fn wfn 4840  cfv 4845  recscrecs 5860  reccrdg 5896
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 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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  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-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-recs 5861  df-irdg 5897
This theorem is referenced by:  rdgivallem  5908
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