Theorem seqomlem2 8440

Description: Lemma for seq_ω. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)

Hypothesis

Ref	Expression
seqomlem.a	⊢ 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)

Assertion

Ref	Expression
seqomlem2	⊢ (𝑄 “ ω) Fn ω

Distinct variable groups:   𝑄,𝑖,𝑣   𝑖,𝐹,𝑣

Allowed substitution hints:   𝐼(𝑣, 𝑖)

Proof of Theorem seqomlem2

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frfnom 8424	. . . . . . 7 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
2		seqomlem.a	. . . . . . . . 9 ⊢ 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)
3	2	reseq1i 5968	. . . . . . . 8 ⊢ (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
4	3	fneq1i 6629	. . . . . . 7 ⊢ ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω)
5	1, 4	mpbir 234	. . . . . 6 ⊢ (𝑄 ↾ ω) Fn ω
6		fvres 6897	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → ((𝑄 ↾ ω)‘𝑏) = (𝑄‘𝑏))
7	2	seqomlem1 8439	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → (𝑄‘𝑏) = ⟨𝑏, (2nd ‘(𝑄‘𝑏))⟩)
8	6, 7	eqtrd 2795	. . . . . . . 8 ⊢ (𝑏 ∈ ω → ((𝑄 ↾ ω)‘𝑏) = ⟨𝑏, (2nd ‘(𝑄‘𝑏))⟩)
9		fvex 6891	. . . . . . . . 9 ⊢ (2nd ‘(𝑄‘𝑏)) ∈ V
10		opelxpi 5692	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ (2nd ‘(𝑄‘𝑏)) ∈ V) → ⟨𝑏, (2nd ‘(𝑄‘𝑏))⟩ ∈ (ω × V))
11	9, 10	mpan2 704	. . . . . . . 8 ⊢ (𝑏 ∈ ω → ⟨𝑏, (2nd ‘(𝑄‘𝑏))⟩ ∈ (ω × V))
12	8, 11	eqeltrd 2860	. . . . . . 7 ⊢ (𝑏 ∈ ω → ((𝑄 ↾ ω)‘𝑏) ∈ (ω × V))
13	12	rgen 3078	. . . . . 6 ⊢ ∀𝑏 ∈ ω ((𝑄 ↾ ω)‘𝑏) ∈ (ω × V)
14		ffnfv 7112	. . . . . 6 ⊢ ((𝑄 ↾ ω):ω⟶(ω × V) ↔ ((𝑄 ↾ ω) Fn ω ∧ ∀𝑏 ∈ ω ((𝑄 ↾ ω)‘𝑏) ∈ (ω × V)))
15	5, 13, 14	mpbir2an 724	. . . . 5 ⊢ (𝑄 ↾ ω):ω⟶(ω × V)
16		frn 6710	. . . . 5 ⊢ ((𝑄 ↾ ω):ω⟶(ω × V) → ran (𝑄 ↾ ω) ⊆ (ω × V))
17	15, 16	ax-mp 5	. . . 4 ⊢ ran (𝑄 ↾ ω) ⊆ (ω × V)
18		df-br 5104	. . . . . . . . . 10 ⊢ (𝑎ran (𝑄 ↾ ω)𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ ran (𝑄 ↾ ω))
19		fvelrnb 6938	. . . . . . . . . . 11 ⊢ ((𝑄 ↾ ω) Fn ω → (⟨𝑎, 𝑏⟩ ∈ ran (𝑄 ↾ ω) ↔ ∃𝑐 ∈ ω ((𝑄 ↾ ω)‘𝑐) = ⟨𝑎, 𝑏⟩))
20	5, 19	ax-mp 5	. . . . . . . . . 10 ⊢ (⟨𝑎, 𝑏⟩ ∈ ran (𝑄 ↾ ω) ↔ ∃𝑐 ∈ ω ((𝑄 ↾ ω)‘𝑐) = ⟨𝑎, 𝑏⟩)
21		fvres 6897	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ ω → ((𝑄 ↾ ω)‘𝑐) = (𝑄‘𝑐))
22	21	eqeq1d 2762	. . . . . . . . . . 11 ⊢ (𝑐 ∈ ω → (((𝑄 ↾ ω)‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ (𝑄‘𝑐) = ⟨𝑎, 𝑏⟩))
23	22	rexbiia 3107	. . . . . . . . . 10 ⊢ (∃𝑐 ∈ ω ((𝑄 ↾ ω)‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ ∃𝑐 ∈ ω (𝑄‘𝑐) = ⟨𝑎, 𝑏⟩)
24	18, 20, 23	3bitri 300	. . . . . . . . 9 ⊢ (𝑎ran (𝑄 ↾ ω)𝑏 ↔ ∃𝑐 ∈ ω (𝑄‘𝑐) = ⟨𝑎, 𝑏⟩)
25	2	seqomlem1 8439	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ ω → (𝑄‘𝑐) = ⟨𝑐, (2nd ‘(𝑄‘𝑐))⟩)
26	25	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → (𝑄‘𝑐) = ⟨𝑐, (2nd ‘(𝑄‘𝑐))⟩)
27	26	eqeq1d 2762	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑐, (2nd ‘(𝑄‘𝑐))⟩ = ⟨𝑎, 𝑏⟩))
28		vex 3454	. . . . . . . . . . . . . . 15 ⊢ 𝑐 ∈ V
29		fvex 6891	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘(𝑄‘𝑐)) ∈ V
30	28, 29	opth1 5451	. . . . . . . . . . . . . 14 ⊢ (⟨𝑐, (2nd ‘(𝑄‘𝑐))⟩ = ⟨𝑎, 𝑏⟩ → 𝑐 = 𝑎)
31	27, 30	biimtrdi 256	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ → 𝑐 = 𝑎))
32		fveqeq2 6887	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑎 → ((𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ (𝑄‘𝑎) = ⟨𝑎, 𝑏⟩))
33	32	biimpd 232	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑎 → ((𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ → (𝑄‘𝑎) = ⟨𝑎, 𝑏⟩))
34	31, 33	syli 40	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ → (𝑄‘𝑎) = ⟨𝑎, 𝑏⟩))
35		fveq2 6878	. . . . . . . . . . . . 13 ⊢ ((𝑄‘𝑎) = ⟨𝑎, 𝑏⟩ → (2nd ‘(𝑄‘𝑎)) = (2nd ‘⟨𝑎, 𝑏⟩))
36		vex 3454	. . . . . . . . . . . . . 14 ⊢ 𝑎 ∈ V
37		vex 3454	. . . . . . . . . . . . . 14 ⊢ 𝑏 ∈ V
38	36, 37	op2nd 7995	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨𝑎, 𝑏⟩) = 𝑏
39	35, 38	eqtr2di 2812	. . . . . . . . . . . 12 ⊢ ((𝑄‘𝑎) = ⟨𝑎, 𝑏⟩ → 𝑏 = (2nd ‘(𝑄‘𝑎)))
40	34, 39	syl6 36	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ → 𝑏 = (2nd ‘(𝑄‘𝑎))))
41	40	rexlimdva 3163	. . . . . . . . . 10 ⊢ (𝑎 ∈ ω → (∃𝑐 ∈ ω (𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ → 𝑏 = (2nd ‘(𝑄‘𝑎))))
42	2	seqomlem1 8439	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ω → (𝑄‘𝑎) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩)
43		fveqeq2 6887	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑎 → ((𝑄‘𝑐) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩ ↔ (𝑄‘𝑎) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩))
44	43	rspcev 3576	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ω ∧ (𝑄‘𝑎) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩) → ∃𝑐 ∈ ω (𝑄‘𝑐) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩)
45	42, 44	mpdan 700	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ω → ∃𝑐 ∈ ω (𝑄‘𝑐) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩)
46		opeq2 4834	. . . . . . . . . . . . 13 ⊢ (𝑏 = (2nd ‘(𝑄‘𝑎)) → ⟨𝑎, 𝑏⟩ = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩)
47	46	eqeq2d 2771	. . . . . . . . . . . 12 ⊢ (𝑏 = (2nd ‘(𝑄‘𝑎)) → ((𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ (𝑄‘𝑐) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩))
48	47	rexbidv 3186	. . . . . . . . . . 11 ⊢ (𝑏 = (2nd ‘(𝑄‘𝑎)) → (∃𝑐 ∈ ω (𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ ∃𝑐 ∈ ω (𝑄‘𝑐) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩))
49	45, 48	syl5ibrcom 250	. . . . . . . . . 10 ⊢ (𝑎 ∈ ω → (𝑏 = (2nd ‘(𝑄‘𝑎)) → ∃𝑐 ∈ ω (𝑄‘𝑐) = ⟨𝑎, 𝑏⟩))
50	41, 49	impbid 215	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → (∃𝑐 ∈ ω (𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ 𝑏 = (2nd ‘(𝑄‘𝑎))))
51	24, 50	bitrid 286	. . . . . . . 8 ⊢ (𝑎 ∈ ω → (𝑎ran (𝑄 ↾ ω)𝑏 ↔ 𝑏 = (2nd ‘(𝑄‘𝑎))))
52	51	alrimiv 1960	. . . . . . 7 ⊢ (𝑎 ∈ ω → ∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏 ↔ 𝑏 = (2nd ‘(𝑄‘𝑎))))
53		fvex 6891	. . . . . . . 8 ⊢ (2nd ‘(𝑄‘𝑎)) ∈ V
54		eqeq2 2772	. . . . . . . . . 10 ⊢ (𝑐 = (2nd ‘(𝑄‘𝑎)) → (𝑏 = 𝑐 ↔ 𝑏 = (2nd ‘(𝑄‘𝑎))))
55	54	bibi2d 345	. . . . . . . . 9 ⊢ (𝑐 = (2nd ‘(𝑄‘𝑎)) → ((𝑎ran (𝑄 ↾ ω)𝑏 ↔ 𝑏 = 𝑐) ↔ (𝑎ran (𝑄 ↾ ω)𝑏 ↔ 𝑏 = (2nd ‘(𝑄‘𝑎)))))
56	55	albidv 1953	. . . . . . . 8 ⊢ (𝑐 = (2nd ‘(𝑄‘𝑎)) → (∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏 ↔ 𝑏 = 𝑐) ↔ ∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏 ↔ 𝑏 = (2nd ‘(𝑄‘𝑎)))))
57	53, 56	spcev 3560	. . . . . . 7 ⊢ (∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏 ↔ 𝑏 = (2nd ‘(𝑄‘𝑎))) → ∃𝑐∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏 ↔ 𝑏 = 𝑐))
58	52, 57	syl 18	. . . . . 6 ⊢ (𝑎 ∈ ω → ∃𝑐∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏 ↔ 𝑏 = 𝑐))
59		eu6 2599	. . . . . 6 ⊢ (∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏 ↔ ∃𝑐∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏 ↔ 𝑏 = 𝑐))
60	58, 59	sylibr 237	. . . . 5 ⊢ (𝑎 ∈ ω → ∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏)
61	60	rgen 3078	. . . 4 ⊢ ∀𝑎 ∈ ω ∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏
62		dff3 7093	. . . 4 ⊢ (ran (𝑄 ↾ ω):ω⟶V ↔ (ran (𝑄 ↾ ω) ⊆ (ω × V) ∧ ∀𝑎 ∈ ω ∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏))
63	17, 61, 62	mpbir2an 724	. . 3 ⊢ ran (𝑄 ↾ ω):ω⟶V
64		df-ima 5668	. . . 4 ⊢ (𝑄 “ ω) = ran (𝑄 ↾ ω)
65	64	feq1i 6693	. . 3 ⊢ ((𝑄 “ ω):ω⟶V ↔ ran (𝑄 ↾ ω):ω⟶V)
66	63, 65	mpbir 234	. 2 ⊢ (𝑄 “ ω):ω⟶V
67		dffn2 6704	. 2 ⊢ ((𝑄 “ ω) Fn ω ↔ (𝑄 “ ω):ω⟶V)
68	66, 67	mpbir 234	1 ⊢ (𝑄 “ ω) Fn ω

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃!weu 2593  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3899  ∅c0 4279  ⟨cop 4590   class class class wbr 5103   I cid 5549   × cxp 5653  ran crn 5656   ↾ cres 5657   “ cima 5658  suc csuc 6359   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415  ωcom 7862  2^nd c2nd 7985  reccrdg 8398

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399

This theorem is used by:  seqomlem3  8441  seqomlem4  8442  fnseqom  8444