Theorem wfrlem14 8046

Description: Lemma for well-ordered recursion. Compute the value of 𝐶. (Contributed by Scott Fenton, 21-Apr-2011.)

Hypotheses

Ref	Expression
wfrlem13.1	⊢ 𝑅 We 𝐴
wfrlem13.2	⊢ 𝑅 Se 𝐴
wfrlem13.3	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
wfrlem13.4	⊢ 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})

Assertion

Ref	Expression
wfrlem14	⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) → (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))

Distinct variable groups:   𝑦,𝐴,𝑧   𝑦,𝐹,𝑧   𝑦,𝐺   𝑦,𝑅,𝑧   𝑦,𝐶

Allowed substitution hints:   𝐶(𝑧)   𝐺(𝑧)

Proof of Theorem wfrlem14

Step	Hyp	Ref	Expression
1		wfrlem13.1	. . 3 ⊢ 𝑅 We 𝐴
2		wfrlem13.2	. . 3 ⊢ 𝑅 Se 𝐴
3		wfrlem13.3	. . 3 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
4		wfrlem13.4	. . 3 ⊢ 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
5	1, 2, 3, 4	wfrlem13 8045	. 2 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))
6		elun 4049	. . . 4 ⊢ (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹 ∨ 𝑦 ∈ {𝑧}))
7		velsn 4543	. . . . 5 ⊢ (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
8	7	orbi2i 913	. . . 4 ⊢ ((𝑦 ∈ dom 𝐹 ∨ 𝑦 ∈ {𝑧}) ↔ (𝑦 ∈ dom 𝐹 ∨ 𝑦 = 𝑧))
9	6, 8	bitri 278	. . 3 ⊢ (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹 ∨ 𝑦 = 𝑧))
10	1, 2, 3	wfrlem12 8044	. . . . . . 7 ⊢ (𝑦 ∈ dom 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
11		fnfun 6457	. . . . . . . 8 ⊢ (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → Fun 𝐶)
12		ssun1 4072	. . . . . . . . . 10 ⊢ 𝐹 ⊆ (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
13	12, 4	sseqtrri 3924	. . . . . . . . 9 ⊢ 𝐹 ⊆ 𝐶
14		funssfv 6716	. . . . . . . . . 10 ⊢ ((Fun 𝐶 ∧ 𝐹 ⊆ 𝐶 ∧ 𝑦 ∈ dom 𝐹) → (𝐶‘𝑦) = (𝐹‘𝑦))
15	3	wfrdmcl 8041	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝐹)
16		fun2ssres 6403	. . . . . . . . . . . 12 ⊢ ((Fun 𝐶 ∧ 𝐹 ⊆ 𝐶 ∧ Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝐹) → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))
17	15, 16	syl3an3 1167	. . . . . . . . . . 11 ⊢ ((Fun 𝐶 ∧ 𝐹 ⊆ 𝐶 ∧ 𝑦 ∈ dom 𝐹) → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))
18	17	fveq2d 6699	. . . . . . . . . 10 ⊢ ((Fun 𝐶 ∧ 𝐹 ⊆ 𝐶 ∧ 𝑦 ∈ dom 𝐹) → (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
19	14, 18	eqeq12d 2752	. . . . . . . . 9 ⊢ ((Fun 𝐶 ∧ 𝐹 ⊆ 𝐶 ∧ 𝑦 ∈ dom 𝐹) → ((𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
20	13, 19	mp3an2 1451	. . . . . . . 8 ⊢ ((Fun 𝐶 ∧ 𝑦 ∈ dom 𝐹) → ((𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
21	11, 20	sylan 583	. . . . . . 7 ⊢ ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ 𝑦 ∈ dom 𝐹) → ((𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
22	10, 21	syl5ibr 249	. . . . . 6 ⊢ ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ 𝑦 ∈ dom 𝐹) → (𝑦 ∈ dom 𝐹 → (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
23	22	ex 416	. . . . 5 ⊢ (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → (𝑦 ∈ dom 𝐹 → (𝑦 ∈ dom 𝐹 → (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))))
24	23	pm2.43d 53	. . . 4 ⊢ (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → (𝑦 ∈ dom 𝐹 → (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
25		vsnid 4564	. . . . . . 7 ⊢ 𝑧 ∈ {𝑧}
26		elun2 4077	. . . . . . 7 ⊢ (𝑧 ∈ {𝑧} → 𝑧 ∈ (dom 𝐹 ∪ {𝑧}))
27	25, 26	ax-mp 5	. . . . . 6 ⊢ 𝑧 ∈ (dom 𝐹 ∪ {𝑧})
28	4	reseq1i 5832	. . . . . . . . . . . . 13 ⊢ (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)) = ((𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ↾ Pred(𝑅, 𝐴, 𝑧))
29		resundir 5851	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ↾ Pred(𝑅, 𝐴, 𝑧)) = ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ({⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧)))
30		wefr 5526	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)
31	1, 30	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ 𝑅 Fr 𝐴
32		predfrirr 6170	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 Fr 𝐴 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑧))
33		ressnop0 6946	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑧) → ({⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧)) = ∅)
34	31, 32, 33	mp2b 10	. . . . . . . . . . . . . . 15 ⊢ ({⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧)) = ∅
35	34	uneq2i 4060	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ({⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧))) = ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ∅)
36		un0 4291	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ∅) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))
37	35, 36	eqtri 2759	. . . . . . . . . . . . 13 ⊢ ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ({⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))
38	28, 29, 37	3eqtri 2763	. . . . . . . . . . . 12 ⊢ (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))
39	38	fveq2i 6698	. . . . . . . . . . 11 ⊢ (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
40	39	opeq2i 4774	. . . . . . . . . 10 ⊢ ⟨𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ = ⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩
41		opex 5333	. . . . . . . . . . 11 ⊢ ⟨𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ V
42	41	elsn 4542	. . . . . . . . . 10 ⊢ (⟨𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↔ ⟨𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ = ⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩)
43	40, 42	mpbir 234	. . . . . . . . 9 ⊢ ⟨𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}
44		elun2 4077	. . . . . . . . 9 ⊢ (⟨𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} → ⟨𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}))
45	43, 44	ax-mp 5	. . . . . . . 8 ⊢ ⟨𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
46	45, 4	eleqtrri 2830	. . . . . . 7 ⊢ ⟨𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ 𝐶
47		fnopfvb 6744	. . . . . . 7 ⊢ ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ 𝑧 ∈ (dom 𝐹 ∪ {𝑧})) → ((𝐶‘𝑧) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))) ↔ ⟨𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ 𝐶))
48	46, 47	mpbiri 261	. . . . . 6 ⊢ ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ 𝑧 ∈ (dom 𝐹 ∪ {𝑧})) → (𝐶‘𝑧) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))))
49	27, 48	mpan2 691	. . . . 5 ⊢ (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → (𝐶‘𝑧) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))))
50		fveq2 6695	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝐶‘𝑦) = (𝐶‘𝑧))
51		predeq3 6144	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
52	51	reseq2d 5836	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))
53	52	fveq2d 6699	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))))
54	50, 53	eqeq12d 2752	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐶‘𝑧) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))))
55	49, 54	syl5ibrcom 250	. . . 4 ⊢ (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → (𝑦 = 𝑧 → (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
56	24, 55	jaod 859	. . 3 ⊢ (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → ((𝑦 ∈ dom 𝐹 ∨ 𝑦 = 𝑧) → (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
57	9, 56	syl5bi 245	. 2 ⊢ (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) → (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
58	5, 57	syl 17	1 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) → (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112   ∖ cdif 3850   ∪ cun 3851   ⊆ wss 3853  ∅c0 4223  {csn 4527  ⟨cop 4533   Fr wfr 5491   Se wse 5492   We wwe 5493  dom cdm 5536   ↾ cres 5538  Predcpred 6139  Fun wfun 6352   Fn wfn 6353  ‘cfv 6358  wrecscwrecs 8024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-po 5453  df-so 5454  df-fr 5494  df-se 5495  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-iota 6316  df-fun 6360  df-fn 6361  df-fv 6366  df-wrecs 8025

This theorem is referenced by:  wfrlem15  8047