Theorem tfrlem6 8371

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 2178, ax-nul 5263, ax-pr 5398, ax-sep 5251 and ax-un 7737. (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 8361	. 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹)
2	1	wfrrel 8320	1 ⊢ Rel recs(𝐹)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  {cab 2738  ∀wral 3076  ∃wrex 3086   E cep 5554   ↾ cres 5657  Rel wrel 5660  Oncon0 6357   Fn wfn 6528  ‘cfv 6533  recscrecs 8360

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 2147  ax-9 2155  ax-11 2194  ax-12 2213  ax-ext 2732

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 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-iota 6489  df-fun 6535  df-fn 6536  df-fv 6541  df-ov 7417  df-frecs 8281  df-wrecs 8312  df-recs 8361

This theorem is used by:  tfrlem7  8373  tfrlem11  8378  tfrlem15  8382  tfrlem16  8383