Theorem tfrlem6 8352

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 5256, ax-pr 5390, ax-sep 5246 and ax-un 7718. (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 8342	. 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹)
2	1	wfrrel 8301	1 ⊢ Rel recs(𝐹)

Syntax hints:   ∧ wa 399   = wceq 1560  {cab 2740  ∀wral 3076  ∃wrex 3086   E cep 5546   ↾ cres 5649  Rel wrel 5652  Oncon0 6346   Fn wfn 6516  ‘cfv 6521  recscrecs 8341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-iota 6477  df-fun 6523  df-fn 6524  df-fv 6529  df-ov 7399  df-frecs 8262  df-wrecs 8293  df-recs 8342

This theorem is referenced by:  tfrlem7  8354  tfrlem11  8359  tfrlem15  8363  tfrlem16  8364