Theorem tfrlem6 8366

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 2175, ax-nul 5268, ax-pr 5403, ax-sep 5256 and ax-un 7734. (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 8356	. 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹)
2	1	wfrrel 8315	1 ⊢ Rel recs(𝐹)

This proof depends on syntax axioms:   ∧ wa 400   = wceq 1569  {cab 2740  ∀wral 3078  ∃wrex 3088   E cep 5559   ↾ cres 5662  Rel wrel 5665  Oncon0 6360   Fn wfn 6531  ‘cfv 6536  recscrecs 8355

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-11 2191  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544  df-ov 7415  df-frecs 8276  df-wrecs 8307  df-recs 8356

This theorem is used by:  tfrlem7  8368  tfrlem11  8373  tfrlem15  8377  tfrlem16  8378