Theorem tfrlem11 2959

Description: Lemma for transfinite recursion. Compute the value of C.

Hypotheses

Ref	Expression
tfrlem.1	⊢ A = {f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))}
tfrlem.2	⊢ F = ∪A
tfrlem.3	⊢ C = (F ∪ {⟨dom F, (G ‘(F ↾ dom F))⟩})

Assertion

Ref	Expression
tfrlem11	⊢ (dom F ∈ On → (y ∈ suc dom F → (C ‘y) = (G ‘(C ↾ y))))

Distinct variable group(s):   x,y,f,A   x,F,y,f   x,G,y,f   x,C,y,f

Proof of Theorem tfrlem11

Step	Hyp	Ref	Expression
1		ssun1 1621	. . . . . . . . 9 ⊢ F ⊆ (F ∪ {⟨dom F, (G ‘(F ↾ dom F))⟩})
2		tfrlem.3	. . . . . . . . 9 ⊢ C = (F ∪ {⟨dom F, (G ‘(F ↾ dom F))⟩})
3	1, 2	sseqtr4 1533	. . . . . . . 8 ⊢ F ⊆ C
4		funssfv 2841	. . . . . . . . . . 11 ⊢ (((Fun C ∧ F ⊆ C) ∧ y ∈ dom F) → (C ‘y) = (F ‘y))
5	4	adantrl 311	. . . . . . . . . 10 ⊢ (((Fun C ∧ F ⊆ C) ∧ (dom F ∈ On ∧ y ∈ dom F)) → (C ‘y) = (F ‘y))
6		fun2ssres 2699	. . . . . . . . . . . 12 ⊢ (((Fun C ∧ F ⊆ C) ∧ y ⊆ dom F) → (C ↾ y) = (F ↾ y))
7	6	fveq2d 2836	. . . . . . . . . . 11 ⊢ (((Fun C ∧ F ⊆ C) ∧ y ⊆ dom F) → (G ‘(C ↾ y)) = (G ‘(F ↾ y)))
8		onelsst 2255	. . . . . . . . . . . 12 ⊢ (dom F ∈ On → (y ∈ dom F → y ⊆ dom F))
9	8	imp 277	. . . . . . . . . . 11 ⊢ ((dom F ∈ On ∧ y ∈ dom F) → y ⊆ dom F)
10	7, 9	sylan2 346	. . . . . . . . . 10 ⊢ (((Fun C ∧ F ⊆ C) ∧ (dom F ∈ On ∧ y ∈ dom F)) → (G ‘(C ↾ y)) = (G ‘(F ↾ y)))
11	5, 10	cleq12d 1115	. . . . . . . . 9 ⊢ (((Fun C ∧ F ⊆ C) ∧ (dom F ∈ On ∧ y ∈ dom F)) → ((C ‘y) = (G ‘(C ↾ y)) ↔ (F ‘y) = (G ‘(F ↾ y))))
12		tfrlem.1	. . . . . . . . . 10 ⊢ A = {f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))}
13		tfrlem.2	. . . . . . . . . 10 ⊢ F = ∪A
14	12, 13	tfrlem9 2957	. . . . . . . . 9 ⊢ (y ∈ dom F → (F ‘y) = (G ‘(F ↾ y)))
15	11, 14	syl5bir 184	. . . . . . . 8 ⊢ (((Fun C ∧ F ⊆ C) ∧ (dom F ∈ On ∧ y ∈ dom F)) → (y ∈ dom F → (C ‘y) = (G ‘(C ↾ y))))
16	3, 15	mpan12 530	. . . . . . 7 ⊢ ((Fun C ∧ (dom F ∈ On ∧ y ∈ dom F)) → (y ∈ dom F → (C ‘y) = (G ‘(C ↾ y))))
17	12, 13, 2	tfrlem10 2958	. . . . . . . 8 ⊢ (dom F ∈ On → C Fn suc dom F)
18		fnfun 2721	. . . . . . . 8 ⊢ (C Fn suc dom F → Fun C)
19	17, 18	syl 12	. . . . . . 7 ⊢ (dom F ∈ On → Fun C)
20	16, 19	sylan 343	. . . . . 6 ⊢ ((dom F ∈ On ∧ (dom F ∈ On ∧ y ∈ dom F)) → (y ∈ dom F → (C ‘y) = (G ‘(C ↾ y))))
21	20	exp32 294	. . . . 5 ⊢ (dom F ∈ On → (dom F ∈ On → (y ∈ dom F → (y ∈ dom F → (C ‘y) = (G ‘(C ↾ y))))))
22	21	pm2.43i 58	. . . 4 ⊢ (dom F ∈ On → (y ∈ dom F → (y ∈ dom F → (C ‘y) = (G ‘(C ↾ y)))))
23	22	pm2.43d 59	. . 3 ⊢ (dom F ∈ On → (y ∈ dom F → (C ‘y) = (G ‘(C ↾ y))))
24		opex 1893	. . . . . . . . 9 ⊢ ⟨y, (G ‘(C ↾ y))⟩ ∈ V
25	24	snid 1830	. . . . . . . 8 ⊢ ⟨y, (G ‘(C ↾ y))⟩ ∈ {⟨y, (G ‘(C ↾ y))⟩}
26		opeq1 1876	. . . . . . . . . . . 12 ⊢ (y = dom F → ⟨y, (G ‘(C ↾ y))⟩ = ⟨dom F, (G ‘(C ↾ y))⟩)
27	26	adantl 305	. . . . . . . . . . 11 ⊢ ((dom F ∈ On ∧ y = dom F) → ⟨y, (G ‘(C ↾ y))⟩ = ⟨dom F, (G ‘(C ↾ y))⟩)
28	3, 6	mpan12 530	. . . . . . . . . . . . . 14 ⊢ ((Fun C ∧ y ⊆ dom F) → (C ↾ y) = (F ↾ y))
29		eqimss 1548	. . . . . . . . . . . . . 14 ⊢ (y = dom F → y ⊆ dom F)
30	28, 19, 29	syl2an 349	. . . . . . . . . . . . 13 ⊢ ((dom F ∈ On ∧ y = dom F) → (C ↾ y) = (F ↾ y))
31		reseq2 2576	. . . . . . . . . . . . . 14 ⊢ (y = dom F → (F ↾ y) = (F ↾ dom F))
32	31	adantl 305	. . . . . . . . . . . . 13 ⊢ ((dom F ∈ On ∧ y = dom F) → (F ↾ y) = (F ↾ dom F))
33	30, 32	eqtrd 1128	. . . . . . . . . . . 12 ⊢ ((dom F ∈ On ∧ y = dom F) → (C ↾ y) = (F ↾ dom F))
34		fveq2 2832	. . . . . . . . . . . 12 ⊢ ((C ↾ y) = (F ↾ dom F) → (G ‘(C ↾ y)) = (G ‘(F ↾ dom F)))
35		opeq2 1877	. . . . . . . . . . . 12 ⊢ ((G ‘(C ↾ y)) = (G ‘(F ↾ dom F)) → ⟨dom F, (G ‘(C ↾ y))⟩ = ⟨dom F, (G ‘(F ↾ dom F))⟩)
36	33, 34, 35	3syl 21	. . . . . . . . . . 11 ⊢ ((dom F ∈ On ∧ y = dom F) → ⟨dom F, (G ‘(C ↾ y))⟩ = ⟨dom F, (G ‘(F ↾ dom F))⟩)
37	27, 36	eqtrd 1128	. . . . . . . . . 10 ⊢ ((dom F ∈ On ∧ y = dom F) → ⟨y, (G ‘(C ↾ y))⟩ = ⟨dom F, (G ‘(F ↾ dom F))⟩)
38	37	sneqd 1818	. . . . . . . . 9 ⊢ ((dom F ∈ On ∧ y = dom F) → {⟨y, (G ‘(C ↾ y))⟩} = {⟨dom F, (G ‘(F ↾ dom F))⟩})
39	38	eleq2d 1156	. . . . . . . 8 ⊢ ((dom F ∈ On ∧ y = dom F) → (⟨y, (G ‘(C ↾ y))⟩ ∈ {⟨y, (G ‘(C ↾ y))⟩} ↔ ⟨y, (G ‘(C ↾ y))⟩ ∈ {⟨dom F, (G ‘(F ↾ dom F))⟩}))
40	25, 39	mpbii 168	. . . . . . 7 ⊢ ((dom F ∈ On ∧ y = dom F) → ⟨y, (G ‘(C ↾ y))⟩ ∈ {⟨dom F, (G ‘(F ↾ dom F))⟩})
41		elun2 1626	. . . . . . 7 ⊢ (⟨y, (G ‘(C ↾ y))⟩ ∈ {⟨dom F, (G ‘(F ↾ dom F))⟩} → ⟨y, (G ‘(C ↾ y))⟩ ∈ (F ∪ {⟨dom F, (G ‘(F ↾ dom F))⟩}))
42	40, 41	syl 12	. . . . . 6 ⊢ ((dom F ∈ On ∧ y = dom F) → ⟨y, (G ‘(C ↾ y))⟩ ∈ (F ∪ {⟨dom F, (G ‘(F ↾ dom F))⟩}))
43	2	eleq2i 1153	. . . . . 6 ⊢ (⟨y, (G ‘(C ↾ y))⟩ ∈ C ↔ ⟨y, (G ‘(C ↾ y))⟩ ∈ (F ∪ {⟨dom F, (G ‘(F ↾ dom F))⟩}))
44	42, 43	sylibr 175	. . . . 5 ⊢ ((dom F ∈ On ∧ y = dom F) → ⟨y, (G ‘(C ↾ y))⟩ ∈ C)
45		fvex 2838	. . . . . . 7 ⊢ (G ‘(C ↾ y)) ∈ V
46	45	fnfvop 2856	. . . . . 6 ⊢ ((C Fn suc dom F ∧ y ∈ suc dom F) → ((C ‘y) = (G ‘(C ↾ y)) ↔ ⟨y, (G ‘(C ↾ y))⟩ ∈ C))
47		visset 1350	. . . . . . 7 ⊢ y ∈ V
48	47	eqelsuc 2307	. . . . . 6 ⊢ (y = dom F → y ∈ suc dom F)
49	46, 17, 48	syl2an 349	. . . . 5 ⊢ ((dom F ∈ On ∧ y = dom F) → ((C ‘y) = (G ‘(C ↾ y)) ↔ ⟨y, (G ‘(C ↾ y))⟩ ∈ C))
50	44, 49	mpbird 171	. . . 4 ⊢ ((dom F ∈ On ∧ y = dom F) → (C ‘y) = (G ‘(C ↾ y)))
51	50	exp 291	. . 3 ⊢ (dom F ∈ On → (y = dom F → (C ‘y) = (G ‘(C ↾ y))))
52	23, 51	jaod 329	. 2 ⊢ (dom F ∈ On → ((y ∈ dom F ∨ y = dom F) → (C ‘y) = (G ‘(C ↾ y))))
53		elsuci 2289	. 2 ⊢ (y ∈ suc dom F → (y ∈ dom F ∨ y = dom F))
54	52, 53	syl5 22	1 ⊢ (dom F ∈ On → (y ∈ suc dom F → (C ‘y) = (G ‘(C ↾ y))))

Syntax hints:   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ∪ cun 1485   ⊆ wss 1487  {csn 1808  ⟨cop 1810  ∪cuni 1919  Oncon0 2199  suc csuc 2201  dom cdm 2410   ↾ cres 2412  Fun wfun 2416   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  tfrlem12 2960

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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