Theorem wfrlem3OLDa 8351

Description: Obsolete version as of 18-Nov-2024. Lemma for well-ordered recursion. Show membership in the class of acceptable functions. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 31-Jul-2020.)

Hypotheses

Ref	Expression
wfrlem1OLD.1	⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
wfrlem3OLDa.2	⊢ 𝐺 ∈ V

Assertion

Ref	Expression
wfrlem3OLDa	⊢ (𝐺 ∈ 𝐵 ↔ ∃𝑧(𝐺 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤)))))

Distinct variable groups:   𝐴,𝑓,𝑤,𝑥,𝑦,𝑧   𝑓,𝐹,𝑤,𝑥,𝑦,𝑧   𝑅,𝑓,𝑤,𝑥,𝑦,𝑧   𝑓,𝐺,𝑥,𝑦,𝑧,𝑤

Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑤,𝑓)

Proof of Theorem wfrlem3OLDa

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wfrlem3OLDa.2	. 2 ⊢ 𝐺 ∈ V
2		fneq1 6659	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 Fn 𝑧 ↔ 𝐺 Fn 𝑧))
3		fveq1 6905	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑤) = (𝐺‘𝑤))
4		reseq1 5991	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑤)))
5	4	fveq2d 6910	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤))))
6	3, 5	eqeq12d 2753	. . . . 5 ⊢ (𝑔 = 𝐺 → ((𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤)))))
7	6	ralbidv 3178	. . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤)))))
8	2, 7	3anbi13d 1440	. . 3 ⊢ (𝑔 = 𝐺 → ((𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) ↔ (𝐺 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤))))))
9	8	exbidv 1921	. 2 ⊢ (𝑔 = 𝐺 → (∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) ↔ ∃𝑧(𝐺 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤))))))
10		wfrlem1OLD.1	. . 3 ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
11	10	wfrlem1OLD 8348	. 2 ⊢ 𝐵 = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}
12	1, 9, 11	elab2 3682	1 ⊢ (𝐺 ∈ 𝐵 ↔ ∃𝑧(𝐺 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑤)))))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2714  ∀wral 3061  Vcvv 3480   ⊆ wss 3951   ↾ cres 5687  Predcpred 6320   Fn wfn 6556  ‘cfv 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569

This theorem is referenced by:  wfrlem17OLD  8365