Theorem tfrlem6 8374

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 5271, ax-pr 5406, ax-sep 5259 and ax-un 7742. (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 8364	. 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹)
2	1	wfrrel 8323	1 ⊢ Rel recs(𝐹)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  {cab 2743  ∀wral 3081  ∃wrex 3091   E cep 5562   ↾ cres 5665  Rel wrel 5668  Oncon0 6364   Fn wfn 6535  ‘cfv 6540  recscrecs 8363

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-iota 6496  df-fun 6542  df-fn 6543  df-fv 6548  df-ov 7422  df-frecs 8284  df-wrecs 8315  df-recs 8364

This theorem is used by:  tfrlem7  8376  tfrlem11  8381  tfrlem15  8385  tfrlem16  8386