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Theorem tfrfun 6268
Description: Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.)
Assertion
Ref Expression
tfrfun Fun recs(𝐹)

Proof of Theorem tfrfun
Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2157 . 2 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
21tfrlem7 6265 1 Fun recs(𝐹)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1335  {cab 2143  wral 2435  wrex 2436  Oncon0 4324  cres 4589  Fun wfun 5165   Fn wfn 5166  cfv 5171  recscrecs 6252
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170  ax-setind 4497
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-id 4254  df-iord 4327  df-on 4329  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-res 4599  df-iota 5136  df-fun 5173  df-fn 5174  df-fv 5179  df-recs 6253
This theorem is referenced by:  tfr1onlembfn  6292  tfr1onlemubacc  6294  tfri1dALT  6299  tfrcllembfn  6305  tfrcllemubacc  6307  tfrcl  6312  frecex  6342  frecfun  6343  frecfcllem  6352  frecsuclem  6354
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