Theorem wfrlem1OLD 8364

Description: Lemma for well-ordered recursion. The final item we are interested in is the union of acceptable functions 𝐵. This lemma just changes bound variables for later use. Obsolete as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 21-Apr-2011.)

Hypothesis

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))))}

Assertion

Ref	Expression
wfrlem1OLD	⊢ 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))))}

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

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

Proof of Theorem wfrlem1OLD

Step	Hyp	Ref	Expression
1		wfrlem1OLD.1	. 2 ⊢ 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))))}
2		fneq1 6670	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑥 ↔ 𝑔 Fn 𝑥))
3		fveq1 6919	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑦) = (𝑔‘𝑦))
4		reseq1 6003	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))
5	4	fveq2d 6924	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))))
6	3, 5	eqeq12d 2756	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))))
7	6	ralbidv 3184	. . . . . 6 ⊢ (𝑓 = 𝑔 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))))
8	2, 7	3anbi13d 1438	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝑔 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))))))
9	8	exbidv 1920	. . . 4 ⊢ (𝑓 = 𝑔 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑥(𝑔 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))))))
10		fneq2 6671	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑔 Fn 𝑥 ↔ 𝑔 Fn 𝑧))
11		sseq1 4034	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ 𝐴 ↔ 𝑧 ⊆ 𝐴))
12		sseq2 4035	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧))
13	12	raleqbi1dv 3346	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧))
14		predeq3 6336	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑤))
15	14	sseq1d 4040	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧 ↔ Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧))
16	15	cbvralvw 3243	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧 ↔ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)
17	13, 16	bitrdi 287	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧))
18	11, 17	anbi12d 631	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ↔ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))
19		raleq 3331	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))))
20		fveq2 6920	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑔‘𝑦) = (𝑔‘𝑤))
21	14	reseq2d 6009	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))
22	21	fveq2d 6924	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))
23	20, 22	eqeq12d 2756	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ((𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
24	23	cbvralvw 3243	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))
25	19, 24	bitrdi 287	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
26	10, 18, 25	3anbi123d 1436	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑔 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))))
27	26	cbvexvw 2036	. . . 4 ⊢ (∃𝑥(𝑔 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
28	9, 27	bitrdi 287	. . 3 ⊢ (𝑓 = 𝑔 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))))
29	28	cbvabv 2815	. 2 ⊢ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}
30	1, 29	eqtri 2768	1 ⊢ 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))))}

Syntax hints:   ∧ wa 395   ∧ w3a 1087   = wceq 1537  ∃wex 1777  {cab 2717  ∀wral 3067   ⊆ wss 3976   ↾ cres 5702  Predcpred 6331   Fn wfn 6568  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581

This theorem is referenced by:  wfrlem2OLD  8365  wfrlem3OLD  8366  wfrlem3OLDa  8367  wfrlem4OLD  8368  wfrdmclOLD  8373