Theorem frrlem6 8221

Description: Lemma for well-founded recursion. The well-founded recursion generator is a relation. (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
frrlem6	⊢ Rel 𝐹

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 frrlem6

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frrlem5.1	. . . . 5 ⊢ 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))))}
2		frrlem5.2	. . . . 5 ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)
3	1, 2	frrlem5 8220	. . . 4 ⊢ 𝐹 = ∪ 𝐵
4	3	releqi 5717	. . 3 ⊢ (Rel 𝐹 ↔ Rel ∪ 𝐵)
5		reluni 5757	. . 3 ⊢ (Rel ∪ 𝐵 ↔ ∀𝑔 ∈ 𝐵 Rel 𝑔)
6	4, 5	bitri 275	. 2 ⊢ (Rel 𝐹 ↔ ∀𝑔 ∈ 𝐵 Rel 𝑔)
7	1	frrlem2 8217	. . 3 ⊢ (𝑔 ∈ 𝐵 → Fun 𝑔)
8		funrel 6498	. . 3 ⊢ (Fun 𝑔 → Rel 𝑔)
9	7, 8	syl 17	. 2 ⊢ (𝑔 ∈ 𝐵 → Rel 𝑔)
10	6, 9	mprgbir 3054	1 ⊢ Rel 𝐹

Syntax hints:   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  ∀wral 3047   ⊆ wss 3897  ∪ cuni 4856   ↾ cres 5616  Rel wrel 5619  Predcpred 6247  Fun wfun 6475   Fn wfn 6476  ‘cfv 6481  (class class class)co 7346  frecscfrecs 8210

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-iota 6437  df-fun 6483  df-fn 6484  df-fv 6489  df-ov 7349  df-frecs 8211

This theorem is referenced by:  frrlem9  8224  frrrel  8236