Theorem tfrlem8 2956

Description: Lemma for transfinite recursion. The domain of F is ordinal.

Hypotheses

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

Assertion

Ref	Expression
tfrlem8	⊢ Ord dom F

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

Proof of Theorem tfrlem8

Step	Hyp	Ref	Expression
1		dftr2 2043	. . 3 ⊢ (Tr dom F ↔ ∀v∀w((v ∈ w ∧ w ∈ dom F) → v ∈ dom F))
2		visset 1350	. . . . . . . . . 10 ⊢ w ∈ V
3	2	eldm2 2528	. . . . . . . . 9 ⊢ (w ∈ dom F ↔ ∃z⟨w, z⟩ ∈ F)
4		tfrlem.2	. . . . . . . . . . . . 13 ⊢ F = ∪A
5		tfrlem.1	. . . . . . . . . . . . . 14 ⊢ A = {f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))}
6	5	unieqi 1928	. . . . . . . . . . . . 13 ⊢ ∪A = ∪{f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))}
7	4, 6	eqtr 1119	. . . . . . . . . . . 12 ⊢ F = ∪{f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))}
8	7	eleq2i 1153	. . . . . . . . . . 11 ⊢ (⟨w, z⟩ ∈ F ↔ ⟨w, z⟩ ∈ ∪{f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))})
9		eluniab 1926	. . . . . . . . . . 11 ⊢ (⟨w, z⟩ ∈ ∪{f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))} ↔ ∃f(⟨w, z⟩ ∈ f ∧ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))))
10	8, 9	bitr 151	. . . . . . . . . 10 ⊢ (⟨w, z⟩ ∈ F ↔ ∃f(⟨w, z⟩ ∈ f ∧ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))))
11	10	biex 733	. . . . . . . . 9 ⊢ (∃z⟨w, z⟩ ∈ F ↔ ∃z∃f(⟨w, z⟩ ∈ f ∧ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))))
12	3, 11	bitr 151	. . . . . . . 8 ⊢ (w ∈ dom F ↔ ∃z∃f(⟨w, z⟩ ∈ f ∧ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))))
13		r19.42v 1303	. . . . . . . . . 10 ⊢ (∃x ∈ On (⟨w, z⟩ ∈ f ∧ (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))) ↔ (⟨w, z⟩ ∈ f ∧ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))))
14		visset 1350	. . . . . . . . . . . . . . . . 17 ⊢ z ∈ V
15	2, 14	fnop 2727	. . . . . . . . . . . . . . . 16 ⊢ ((f Fn x ∧ ⟨w, z⟩ ∈ f) → w ∈ x)
16	15	ancoms 334	. . . . . . . . . . . . . . 15 ⊢ ((⟨w, z⟩ ∈ f ∧ f Fn x) → w ∈ x)
17	16	adantrr 312	. . . . . . . . . . . . . 14 ⊢ ((⟨w, z⟩ ∈ f ∧ (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))) → w ∈ x)
18	17	adantl 305	. . . . . . . . . . . . 13 ⊢ ((x ∈ On ∧ (⟨w, z⟩ ∈ f ∧ (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y))))) → w ∈ x)
19		ra4e 1244	. . . . . . . . . . . . . . . 16 ⊢ ((x ∈ On ∧ (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))) → ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y))))
20	5	cleqabi 1176	. . . . . . . . . . . . . . . . 17 ⊢ (f ∈ A ↔ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y))))
21		elssuni 1940	. . . . . . . . . . . . . . . . . 18 ⊢ (f ∈ A → f ⊆ ∪A)
22	4	sseq2i 1525	. . . . . . . . . . . . . . . . . . 19 ⊢ (f ⊆ F ↔ f ⊆ ∪A)
23		dmss 2530	. . . . . . . . . . . . . . . . . . 19 ⊢ (f ⊆ F → dom f ⊆ dom F)
24	22, 23	sylbir 176	. . . . . . . . . . . . . . . . . 18 ⊢ (f ⊆ ∪A → dom f ⊆ dom F)
25	21, 24	syl 12	. . . . . . . . . . . . . . . . 17 ⊢ (f ∈ A → dom f ⊆ dom F)
26	20, 25	sylbir 176	. . . . . . . . . . . . . . . 16 ⊢ (∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y))) → dom f ⊆ dom F)
27	19, 26	syl 12	. . . . . . . . . . . . . . 15 ⊢ ((x ∈ On ∧ (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))) → dom f ⊆ dom F)
28		fndm 2723	. . . . . . . . . . . . . . . . 17 ⊢ (f Fn x → dom f = x)
29	28	sseq1d 1527	. . . . . . . . . . . . . . . 16 ⊢ (f Fn x → (dom f ⊆ dom F ↔ x ⊆ dom F))
30	29	ad2antrl 322	. . . . . . . . . . . . . . 15 ⊢ ((x ∈ On ∧ (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))) → (dom f ⊆ dom F ↔ x ⊆ dom F))
31	27, 30	mpbid 170	. . . . . . . . . . . . . 14 ⊢ ((x ∈ On ∧ (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))) → x ⊆ dom F)
32	31	adantrl 311	. . . . . . . . . . . . 13 ⊢ ((x ∈ On ∧ (⟨w, z⟩ ∈ f ∧ (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y))))) → x ⊆ dom F)
33	18, 32	jca 236	. . . . . . . . . . . 12 ⊢ ((x ∈ On ∧ (⟨w, z⟩ ∈ f ∧ (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y))))) → (w ∈ x ∧ x ⊆ dom F))
34	33	exp 291	. . . . . . . . . . 11 ⊢ (x ∈ On → ((⟨w, z⟩ ∈ f ∧ (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))) → (w ∈ x ∧ x ⊆ dom F)))
35	34	r19.22i 1273	. . . . . . . . . 10 ⊢ (∃x ∈ On (⟨w, z⟩ ∈ f ∧ (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))) → ∃x ∈ On (w ∈ x ∧ x ⊆ dom F))
36	13, 35	sylbir 176	. . . . . . . . 9 ⊢ ((⟨w, z⟩ ∈ f ∧ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))) → ∃x ∈ On (w ∈ x ∧ x ⊆ dom F))
37	36	19.23aivv 953	. . . . . . . 8 ⊢ (∃z∃f(⟨w, z⟩ ∈ f ∧ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))) → ∃x ∈ On (w ∈ x ∧ x ⊆ dom F))
38	12, 37	sylbi 174	. . . . . . 7 ⊢ (w ∈ dom F → ∃x ∈ On (w ∈ x ∧ x ⊆ dom F))
39	38	anim2i 270	. . . . . 6 ⊢ ((v ∈ w ∧ w ∈ dom F) → (v ∈ w ∧ ∃x ∈ On (w ∈ x ∧ x ⊆ dom F)))
40		r19.42v 1303	. . . . . 6 ⊢ (∃x ∈ On (v ∈ w ∧ (w ∈ x ∧ x ⊆ dom F)) ↔ (v ∈ w ∧ ∃x ∈ On (w ∈ x ∧ x ⊆ dom F)))
41	39, 40	sylibr 175	. . . . 5 ⊢ ((v ∈ w ∧ w ∈ dom F) → ∃x ∈ On (v ∈ w ∧ (w ∈ x ∧ x ⊆ dom F)))
42		ontr1 2258	. . . . . . . . . 10 ⊢ (x ∈ On → ((v ∈ w ∧ w ∈ x) → v ∈ x))
43		ssel 1502	. . . . . . . . . 10 ⊢ (x ⊆ dom F → (v ∈ x → v ∈ dom F))
44	42, 43	sylan9r 360	. . . . . . . . 9 ⊢ ((x ⊆ dom F ∧ x ∈ On) → ((v ∈ w ∧ w ∈ x) → v ∈ dom F))
45	44	exp4b 296	. . . . . . . 8 ⊢ (x ⊆ dom F → (x ∈ On → (v ∈ w → (w ∈ x → v ∈ dom F))))
46	45	com4l 39	. . . . . . 7 ⊢ (x ∈ On → (v ∈ w → (w ∈ x → (x ⊆ dom F → v ∈ dom F))))
47	46	imp4d 285	. . . . . 6 ⊢ (x ∈ On → ((v ∈ w ∧ (w ∈ x ∧ x ⊆ dom F)) → v ∈ dom F))
48	47	r19.23aiv 1284	. . . . 5 ⊢ (∃x ∈ On (v ∈ w ∧ (w ∈ x ∧ x ⊆ dom F)) → v ∈ dom F)
49	41, 48	syl 12	. . . 4 ⊢ ((v ∈ w ∧ w ∈ dom F) → v ∈ dom F)
50	49	ax-gen 677	. . 3 ⊢ ∀w((v ∈ w ∧ w ∈ dom F) → v ∈ dom F)
51	1, 50	mpgbir 686	. 2 ⊢ Tr dom F
52		onelon 2223	. . . . . . . . . . . . . 14 ⊢ ((x ∈ On ∧ w ∈ x) → w ∈ On)
53	52, 15	sylan2 346	. . . . . . . . . . . . 13 ⊢ ((x ∈ On ∧ (f Fn x ∧ ⟨w, z⟩ ∈ f)) → w ∈ On)
54	53	exp32 294	. . . . . . . . . . . 12 ⊢ (x ∈ On → (f Fn x → (⟨w, z⟩ ∈ f → w ∈ On)))
55	54	com12 13	. . . . . . . . . . 11 ⊢ (f Fn x → (x ∈ On → (⟨w, z⟩ ∈ f → w ∈ On)))
56	55	adantr 306	. . . . . . . . . 10 ⊢ ((f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y))) → (x ∈ On → (⟨w, z⟩ ∈ f → w ∈ On)))
57	56	com13 33	. . . . . . . . 9 ⊢ (⟨w, z⟩ ∈ f → (x ∈ On → ((f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y))) → w ∈ On)))
58	57	r19.23adv 1286	. . . . . . . 8 ⊢ (⟨w, z⟩ ∈ f → (∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y))) → w ∈ On))
59	58	imp 277	. . . . . . 7 ⊢ ((⟨w, z⟩ ∈ f ∧ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))) → w ∈ On)
60	59	19.23aiv 952	. . . . . 6 ⊢ (∃f(⟨w, z⟩ ∈ f ∧ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))) → w ∈ On)
61	10, 60	sylbi 174	. . . . 5 ⊢ (⟨w, z⟩ ∈ F → w ∈ On)
62	61	19.23aiv 952	. . . 4 ⊢ (∃z⟨w, z⟩ ∈ F → w ∈ On)
63	3, 62	sylbi 174	. . 3 ⊢ (w ∈ dom F → w ∈ On)
64	63	ssriv 1508	. 2 ⊢ dom F ⊆ On
65		ordon 2238	. 2 ⊢ Ord On
66		trssord 2216	. 2 ⊢ ((Tr dom F ∧ dom F ⊆ On ∧ Ord On) → Ord dom F)
67	51, 64, 65, 66	mp3an 642	1 ⊢ Ord dom F

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ⊆ wss 1487  ⟨cop 1810  ∪cuni 1919  Tr wtr 2041  Ord word 2198  Oncon0 2199  dom cdm 2410   ↾ cres 2412   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  tfrlem13 2961

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-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  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-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-dm 2428  df-fn 2433

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