Theorem tfrlem6 8400

Description: Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.) Avoid ax-10 2179, ax-nul 5262, ax-pr 5394, ax-sep 5251 and ax-un 7758. (Revised by Umit Teoman Dogan, 10-Jun-2026.)

Hypothesis

Ref	Expression
tfrlem.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

Assertion

Ref	Expression
tfrlem6	⊢ Rel recs(𝐹)

Distinct variable group:   𝑥,𝑓,𝑦,𝐹

Allowed substitution hints:   𝐴(𝑥, 𝑦, 𝑓)

Proof of Theorem tfrlem6

Step	Hyp	Ref	Expression
1		df-recs 8390	. 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹)
2	1	wfrrel 8349	1 ⊢ Rel recs(𝐹)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569  {cab 2741  ∀wral 3079  ∃wrex 3089   E cep 5553   ↾ cres 5656  Rel wrel 5659  Oncon0 6368   Fn wfn 6541  ‘cfv 6546  recscrecs 8389

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-11 2196  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-iota 6500  df-fun 6548  df-fn 6549  df-fv 6554  df-ov 7430  df-frecs 8310  df-wrecs 8341  df-recs 8390

This theorem is used by:  tfrlem7  8402  tfrlem11  8407  tfrlem15  8411  tfrlem16  8412