Theorem frr2 9518

Description: Law of general well-founded recursion, part two. Now we establish the value of 𝐹 within 𝐴. (Contributed by Scott Fenton, 11-Sep-2023.)

Hypothesis

Ref	Expression
frr.1	⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)

Assertion

Ref	Expression
frr2	⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

Proof of Theorem frr2

Dummy variables 𝑥 𝑦 𝑢 𝑣 𝑎 𝑏 𝑐 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frr.1	. . . . . 6 ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)
2	1	frr1 9517	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Fn 𝐴)
3	2	fndmd 6538	. . . 4 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → dom 𝐹 = 𝐴)
4	3	eleq2d 2824	. . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝐴))
5	4	pm5.32i 575	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) ↔ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ 𝐴))
6		fveq2 6774	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋))
7		id 22	. . . . . . 7 ⊢ (𝑦 = 𝑋 → 𝑦 = 𝑋)
8		predeq3 6206	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑋))
9	8	reseq2d 5891	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))
10	7, 9	oveq12d 7293	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
11	6, 10	eqeq12d 2754	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))))
12	11	imbi2d 341	. . . 4 ⊢ (𝑦 = 𝑋 → (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))))
13		eqid 2738	. . . . . . 7 ⊢ {𝑎 ∣ ∃𝑏(𝑎 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑐 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑐) ⊆ 𝑏) ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝑐𝐺(𝑎 ↾ Pred(𝑅, 𝐴, 𝑐))))} = {𝑎 ∣ ∃𝑏(𝑎 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑐 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑐) ⊆ 𝑏) ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝑐𝐺(𝑎 ↾ Pred(𝑅, 𝐴, 𝑐))))}
14	13	frrlem1 8102	. . . . . 6 ⊢ {𝑎 ∣ ∃𝑏(𝑎 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑐 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑐) ⊆ 𝑏) ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝑐𝐺(𝑎 ↾ Pred(𝑅, 𝐴, 𝑐))))} = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
15	14, 1	frrlem15 9515	. . . . . 6 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ {𝑎 ∣ ∃𝑏(𝑎 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑐 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑐) ⊆ 𝑏) ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝑐𝐺(𝑎 ↾ Pred(𝑅, 𝐴, 𝑐))))} ∧ ℎ ∈ {𝑎 ∣ ∃𝑏(𝑎 Fn 𝑏 ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑐 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑐) ⊆ 𝑏) ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝑐𝐺(𝑎 ↾ Pred(𝑅, 𝐴, 𝑐))))})) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
16	14, 1, 15	frrlem10 8111	. . . . 5 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
17	16	expcom 414	. . . 4 ⊢ (𝑦 ∈ dom 𝐹 → ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (𝐹‘𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
18	12, 17	vtoclga 3513	. . 3 ⊢ (𝑋 ∈ dom 𝐹 → ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))))
19	18	impcom 408	. 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
20	5, 19	sylbir 234	1 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715  ∀wral 3064   ⊆ wss 3887   Fr wfr 5541   Se wse 5542  dom cdm 5589   ↾ cres 5591  Predcpred 6201   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275  frecscfrecs 8096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-ttrcl 9466

This theorem is referenced by:  frr3  9519