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Theorem tfrfun 6210
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 2137 . 2 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
21tfrlem7 6207 1 Fun recs(𝐹)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1331  {cab 2123  wral 2414  wrex 2415  Oncon0 4280  cres 4536  Fun wfun 5112   Fn wfn 5113  cfv 5118  recscrecs 6194
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-setind 4447
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-res 4546  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126  df-recs 6195
This theorem is referenced by:  tfr1onlembfn  6234  tfr1onlemubacc  6236  tfri1dALT  6241  tfrcllembfn  6247  tfrcllemubacc  6249  tfrcl  6254  frecex  6284  frecfun  6285  frecfcllem  6294  frecsuclem  6296
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