Theorem wfrlem1 7948

Description: Lemma for well-founded recursion. The final item we are interested in is the union of acceptable functions 𝐵. This lemma just changes bound variables for later use. (Contributed by Scott Fenton, 21-Apr-2011.)

Hypothesis

Ref	Expression
wfrlem1.1	⊢ 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
wfrlem1	⊢ 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 wfrlem1

Step	Hyp	Ref	Expression
1		wfrlem1.1	. 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 6438	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑥 ↔ 𝑔 Fn 𝑥))
3		fveq1 6663	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑦) = (𝑔‘𝑦))
4		reseq1 5841	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))
5	4	fveq2d 6668	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))))
6	3, 5	eqeq12d 2837	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)))))
7	6	ralbidv 3197	. . . . . 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 1434	. . . . 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 1918	. . . 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 6439	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑔 Fn 𝑥 ↔ 𝑔 Fn 𝑧))
11		sseq1 3991	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 ⊆ 𝐴 ↔ 𝑧 ⊆ 𝐴))
12		sseq2 3992	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧))
13	12	raleqbi1dv 3403	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧))
14		predeq3 6146	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑤))
15	14	sseq1d 3997	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧 ↔ Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧))
16	15	cbvralvw 3449	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑧 ↔ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)
17	13, 16	syl6bb 289	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧))
18	11, 17	anbi12d 632	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ↔ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)))
19		raleq 3405	. . . . . . 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 6664	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑔‘𝑦) = (𝑔‘𝑤))
21	14	reseq2d 5847	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑔 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))
22	21	fveq2d 6668	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))
23	20, 22	eqeq12d 2837	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ((𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
24	23	cbvralvw 3449	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))
25	19, 24	syl6bb 289	. . . . . 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 1432	. . . . 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 2040	. . . 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	syl6bb 289	. . 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 2889	. 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 2844	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 398   ∧ w3a 1083   = wceq 1533  ∃wex 1776  {cab 2799  ∀wral 3138   ⊆ wss 3935   ↾ cres 5551  Predcpred 6141   Fn wfn 6344  ‘cfv 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-iota 6308  df-fun 6351  df-fn 6352  df-fv 6357

This theorem is referenced by:  wfrlem2  7949  wfrlem3  7950  wfrlem3a  7951  wfrlem4  7952  wfrdmcl  7957