Theorem tfrfun 6344

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.

Step	Hyp	Ref	Expression
1		eqid 2189	. 2 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2	1	tfrlem7 6341	1 ⊢ Fun recs(𝐹)

Syntax hints:   ∧ wa 104   = wceq 1364  {cab 2175  ∀wral 2468  ∃wrex 2469  Oncon0 4381   ↾ cres 4646  Fun wfun 5229   Fn wfn 5230  ‘cfv 5235  recscrecs 6328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-setind 4554

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-recs 6329

This theorem is referenced by:  tfr1onlembfn  6368  tfr1onlemubacc  6370  tfri1dALT  6375  tfrcllembfn  6381  tfrcllemubacc  6383  tfrcl  6388  frecex  6418  frecfun  6419  frecfcllem  6428  frecsuclem  6430