Theorem frrlem5 8168

Description: Lemma for well-founded recursion. State the well-founded recursion generator in terms of the acceptable functions. (Contributed by Scott Fenton, 27-Aug-2022.)

Hypotheses

Ref	Expression
frrlem5.1	⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
frrlem5.2	⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)

Assertion

Ref	Expression
frrlem5	⊢ 𝐹 = ∪ 𝐵

Distinct variable groups:   𝐴,𝑓,𝑥,𝑦   𝑓,𝐺,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦,𝑓)   𝐹(𝑥,𝑦,𝑓)

Proof of Theorem frrlem5

Step	Hyp	Ref	Expression
1		df-frecs 8159	. 2 ⊢ frecs(𝑅, 𝐴, 𝐺) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
2		frrlem5.2	. 2 ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)
3		frrlem5.1	. . 3 ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
4	3	unieqi 4864	. 2 ⊢ ∪ 𝐵 = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
5	1, 2, 4	3eqtr4i 2774	1 ⊢ 𝐹 = ∪ 𝐵

Syntax hints:   ∧ wa 396   ∧ w3a 1086   = wceq 1540  ∃wex 1780  {cab 2713  ∀wral 3061   ⊆ wss 3897  ∪ cuni 4851   ↾ cres 5616  Predcpred 6231   Fn wfn 6468  ‘cfv 6473  (class class class)co 7329  frecscfrecs 8158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443  df-in 3904  df-ss 3914  df-uni 4852  df-frecs 8159

This theorem is referenced by:  frrlem6  8169  frrlem7  8170  frrlem8  8171  frrlem9  8172  frrlem10  8173  frrlem14  8177