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Theorem rdgfun 6238
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 2117 . . 3 {𝑓 ∣ ∃𝑦 ∈ On (𝑓 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑓𝑧) = ((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))‘(𝑓𝑧)))} = {𝑓 ∣ ∃𝑦 ∈ On (𝑓 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑓𝑧) = ((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))‘(𝑓𝑧)))}
21tfrlem7 6182 . 2 Fun recs((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))))
3 df-irdg 6235 . . 3 rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥)))))
43funeqi 5114 . 2 (Fun rec(𝐹, 𝐴) ↔ Fun recs((𝑔 ∈ V ↦ (𝐴 𝑥 ∈ dom 𝑔(𝐹‘(𝑔𝑥))))))
52, 4mpbir 145 1 Fun rec(𝐹, 𝐴)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1316  {cab 2103  wral 2393  wrex 2394  Vcvv 2660  cun 3039   ciun 3783  cmpt 3959  Oncon0 4255  dom cdm 4509  cres 4511  Fun wfun 5087   Fn wfn 5088  cfv 5093  recscrecs 6169  reccrdg 6234
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-irdg 6235
This theorem is referenced by:  rdgivallem  6246
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