Theorem wfrlem14OLD 8341

Description: Obsolete version as of 18-Nov-2024. Lemma for well-ordered recursion. Compute the value of 𝐶. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 21-Apr-2011.)

Hypotheses

Ref	Expression
wfrlem13OLD.1	⊢ 𝑅 We 𝐴
wfrlem13OLD.2	⊢ 𝑅 Se 𝐴
wfrlem13OLD.3	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
wfrlem13OLD.4	⊢ 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})

Assertion

Ref	Expression
wfrlem14OLD	⊢ (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 wfrlem14OLD

Step	Hyp	Ref	Expression
1		wfrlem13OLD.1	. . 3 ⊢ 𝑅 We 𝐴
2		wfrlem13OLD.2	. . 3 ⊢ 𝑅 Se 𝐴
3		wfrlem13OLD.3	. . 3 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
4		wfrlem13OLD.4	. . 3 ⊢ 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
5	1, 2, 3, 4	wfrlem13OLD 8340	. 2 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))
6		elun 4133	. . . 4 ⊢ (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹 ∨ 𝑦 ∈ {𝑧}))
7		velsn 4622	. . . . 5 ⊢ (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
8	7	orbi2i 912	. . . 4 ⊢ ((𝑦 ∈ dom 𝐹 ∨ 𝑦 ∈ {𝑧}) ↔ (𝑦 ∈ dom 𝐹 ∨ 𝑦 = 𝑧))
9	6, 8	bitri 275	. . 3 ⊢ (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹 ∨ 𝑦 = 𝑧))
10	1, 2, 3	wfrlem12OLD 8339	. . . . . . 7 ⊢ (𝑦 ∈ dom 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
11		fnfun 6643	. . . . . . . 8 ⊢ (𝐶 Fn (dom 𝐹 ∪ {𝑧}) → Fun 𝐶)
12		ssun1 4158	. . . . . . . . . 10 ⊢ 𝐹 ⊆ (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
13	12, 4	sseqtrri 4013	. . . . . . . . 9 ⊢ 𝐹 ⊆ 𝐶
14		funssfv 6902	. . . . . . . . . 10 ⊢ ((Fun 𝐶 ∧ 𝐹 ⊆ 𝐶 ∧ 𝑦 ∈ dom 𝐹) → (𝐶‘𝑦) = (𝐹‘𝑦))
15	3	wfrdmclOLD 8336	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝐹)
16		fun2ssres 6586	. . . . . . . . . . . 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 1165	. . . . . . . . . . 11 ⊢ ((Fun 𝐶 ∧ 𝐹 ⊆ 𝐶 ∧ 𝑦 ∈ dom 𝐹) → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))
18	17	fveq2d 6885	. . . . . . . . . 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 580	. . . . . . 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	imbitrrid 246	. . . . . 6 ⊢ ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ 𝑦 ∈ dom 𝐹) → (𝑦 ∈ dom 𝐹 → (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
23	22	ex 412	. . . . 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 4644	. . . . . . 7 ⊢ 𝑧 ∈ {𝑧}
26		elun2 4163	. . . . . . 7 ⊢ (𝑧 ∈ {𝑧} → 𝑧 ∈ (dom 𝐹 ∪ {𝑧}))
27	25, 26	ax-mp 5	. . . . . 6 ⊢ 𝑧 ∈ (dom 𝐹 ∪ {𝑧})
28	4	reseq1i 5967	. . . . . . . . . . . . 13 ⊢ (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)) = ((𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) ↾ Pred(𝑅, 𝐴, 𝑧))
29		resundir 5986	. . . . . . . . . . . . 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 5649	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)
31	1, 30	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ 𝑅 Fr 𝐴
32		predfrirr 6328	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 Fr 𝐴 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑧))
33		ressnop0 7148	. . . . . . . . . . . . . . . 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 4145	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ({⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩} ↾ Pred(𝑅, 𝐴, 𝑧))) = ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ ∅)
36		un0 4374	. . . . . . . . . . . . . 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 6884	. . . . . . . . . . 11 ⊢ (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
40	39	opeq2i 4858	. . . . . . . . . 10 ⊢ ⟨𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ = ⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩
41		opex 5444	. . . . . . . . . . 11 ⊢ ⟨𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ V
42	41	elsn 4621	. . . . . . . . . 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 231	. . . . . . . . 9 ⊢ ⟨𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}
44		elun2 4163	. . . . . . . . 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 2834	. . . . . . 7 ⊢ ⟨𝑧, (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩ ∈ 𝐶
47		fnopfvb 6935	. . . . . . 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 258	. . . . . 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 6881	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝐶‘𝑦) = (𝐶‘𝑧))
51		predeq3 6299	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
52	51	reseq2d 5971	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐶 ↾ Pred(𝑅, 𝐴, 𝑧)))
53	52	fveq2d 6885	. . . . . 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 247	. . . 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	biimtrid 242	. 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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∖ cdif 3928   ∪ cun 3929   ⊆ wss 3931  ∅c0 4313  {csn 4606  ⟨cop 4612   Fr wfr 5608   Se wse 5609   We wwe 5610  dom cdm 5659   ↾ cres 5661  Predcpred 6294  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536  wrecscwrecs 8315

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-ov 7413  df-2nd 7994  df-frecs 8285  df-wrecs 8316

This theorem is referenced by:  wfrlem15OLD  8342