Theorem seqomlem2 8383

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 8367	. . . . . . 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 5934	. . . . . . . 8 ⊢ (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
4	3	fneq1i 6589	. . . . . . 7 ⊢ ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω)
5	1, 4	mpbir 231	. . . . . 6 ⊢ (𝑄 ↾ ω) Fn ω
6		fvres 6853	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → ((𝑄 ↾ ω)‘𝑏) = (𝑄‘𝑏))
7	2	seqomlem1 8382	. . . . . . . . 9 ⊢ (𝑏 ∈ ω → (𝑄‘𝑏) = ⟨𝑏, (2nd ‘(𝑄‘𝑏))⟩)
8	6, 7	eqtrd 2772	. . . . . . . 8 ⊢ (𝑏 ∈ ω → ((𝑄 ↾ ω)‘𝑏) = ⟨𝑏, (2nd ‘(𝑄‘𝑏))⟩)
9		fvex 6847	. . . . . . . . 9 ⊢ (2nd ‘(𝑄‘𝑏)) ∈ V
10		opelxpi 5661	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ (2nd ‘(𝑄‘𝑏)) ∈ V) → ⟨𝑏, (2nd ‘(𝑄‘𝑏))⟩ ∈ (ω × V))
11	9, 10	mpan2 692	. . . . . . . 8 ⊢ (𝑏 ∈ ω → ⟨𝑏, (2nd ‘(𝑄‘𝑏))⟩ ∈ (ω × V))
12	8, 11	eqeltrd 2837	. . . . . . 7 ⊢ (𝑏 ∈ ω → ((𝑄 ↾ ω)‘𝑏) ∈ (ω × V))
13	12	rgen 3054	. . . . . 6 ⊢ ∀𝑏 ∈ ω ((𝑄 ↾ ω)‘𝑏) ∈ (ω × V)
14		ffnfv 7065	. . . . . 6 ⊢ ((𝑄 ↾ ω):ω⟶(ω × V) ↔ ((𝑄 ↾ ω) Fn ω ∧ ∀𝑏 ∈ ω ((𝑄 ↾ ω)‘𝑏) ∈ (ω × V)))
15	5, 13, 14	mpbir2an 712	. . . . 5 ⊢ (𝑄 ↾ ω):ω⟶(ω × V)
16		frn 6669	. . . . 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 5087	. . . . . . . . . 10 ⊢ (𝑎ran (𝑄 ↾ ω)𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ ran (𝑄 ↾ ω))
19		fvelrnb 6894	. . . . . . . . . . 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 6853	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ ω → ((𝑄 ↾ ω)‘𝑐) = (𝑄‘𝑐))
22	21	eqeq1d 2739	. . . . . . . . . . 11 ⊢ (𝑐 ∈ ω → (((𝑄 ↾ ω)‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ (𝑄‘𝑐) = ⟨𝑎, 𝑏⟩))
23	22	rexbiia 3083	. . . . . . . . . 10 ⊢ (∃𝑐 ∈ ω ((𝑄 ↾ ω)‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ ∃𝑐 ∈ ω (𝑄‘𝑐) = ⟨𝑎, 𝑏⟩)
24	18, 20, 23	3bitri 297	. . . . . . . . 9 ⊢ (𝑎ran (𝑄 ↾ ω)𝑏 ↔ ∃𝑐 ∈ ω (𝑄‘𝑐) = ⟨𝑎, 𝑏⟩)
25	2	seqomlem1 8382	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ ω → (𝑄‘𝑐) = ⟨𝑐, (2nd ‘(𝑄‘𝑐))⟩)
26	25	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → (𝑄‘𝑐) = ⟨𝑐, (2nd ‘(𝑄‘𝑐))⟩)
27	26	eqeq1d 2739	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑐, (2nd ‘(𝑄‘𝑐))⟩ = ⟨𝑎, 𝑏⟩))
28		vex 3434	. . . . . . . . . . . . . . 15 ⊢ 𝑐 ∈ V
29		fvex 6847	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘(𝑄‘𝑐)) ∈ V
30	28, 29	opth1 5423	. . . . . . . . . . . . . 14 ⊢ (⟨𝑐, (2nd ‘(𝑄‘𝑐))⟩ = ⟨𝑎, 𝑏⟩ → 𝑐 = 𝑎)
31	27, 30	biimtrdi 253	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ → 𝑐 = 𝑎))
32		fveqeq2 6843	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑎 → ((𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ (𝑄‘𝑎) = ⟨𝑎, 𝑏⟩))
33	32	biimpd 229	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑎 → ((𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ → (𝑄‘𝑎) = ⟨𝑎, 𝑏⟩))
34	31, 33	syli 39	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ → (𝑄‘𝑎) = ⟨𝑎, 𝑏⟩))
35		fveq2 6834	. . . . . . . . . . . . 13 ⊢ ((𝑄‘𝑎) = ⟨𝑎, 𝑏⟩ → (2nd ‘(𝑄‘𝑎)) = (2nd ‘⟨𝑎, 𝑏⟩))
36		vex 3434	. . . . . . . . . . . . . 14 ⊢ 𝑎 ∈ V
37		vex 3434	. . . . . . . . . . . . . 14 ⊢ 𝑏 ∈ V
38	36, 37	op2nd 7944	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨𝑎, 𝑏⟩) = 𝑏
39	35, 38	eqtr2di 2789	. . . . . . . . . . . 12 ⊢ ((𝑄‘𝑎) = ⟨𝑎, 𝑏⟩ → 𝑏 = (2nd ‘(𝑄‘𝑎)))
40	34, 39	syl6 35	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ → 𝑏 = (2nd ‘(𝑄‘𝑎))))
41	40	rexlimdva 3139	. . . . . . . . . 10 ⊢ (𝑎 ∈ ω → (∃𝑐 ∈ ω (𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ → 𝑏 = (2nd ‘(𝑄‘𝑎))))
42	2	seqomlem1 8382	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ω → (𝑄‘𝑎) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩)
43		fveqeq2 6843	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑎 → ((𝑄‘𝑐) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩ ↔ (𝑄‘𝑎) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩))
44	43	rspcev 3565	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ω ∧ (𝑄‘𝑎) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩) → ∃𝑐 ∈ ω (𝑄‘𝑐) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩)
45	42, 44	mpdan 688	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ω → ∃𝑐 ∈ ω (𝑄‘𝑐) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩)
46		opeq2 4818	. . . . . . . . . . . . 13 ⊢ (𝑏 = (2nd ‘(𝑄‘𝑎)) → ⟨𝑎, 𝑏⟩ = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩)
47	46	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ (𝑏 = (2nd ‘(𝑄‘𝑎)) → ((𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ (𝑄‘𝑐) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩))
48	47	rexbidv 3162	. . . . . . . . . . 11 ⊢ (𝑏 = (2nd ‘(𝑄‘𝑎)) → (∃𝑐 ∈ ω (𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ ∃𝑐 ∈ ω (𝑄‘𝑐) = ⟨𝑎, (2nd ‘(𝑄‘𝑎))⟩))
49	45, 48	syl5ibrcom 247	. . . . . . . . . 10 ⊢ (𝑎 ∈ ω → (𝑏 = (2nd ‘(𝑄‘𝑎)) → ∃𝑐 ∈ ω (𝑄‘𝑐) = ⟨𝑎, 𝑏⟩))
50	41, 49	impbid 212	. . . . . . . . 9 ⊢ (𝑎 ∈ ω → (∃𝑐 ∈ ω (𝑄‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ 𝑏 = (2nd ‘(𝑄‘𝑎))))
51	24, 50	bitrid 283	. . . . . . . 8 ⊢ (𝑎 ∈ ω → (𝑎ran (𝑄 ↾ ω)𝑏 ↔ 𝑏 = (2nd ‘(𝑄‘𝑎))))
52	51	alrimiv 1929	. . . . . . 7 ⊢ (𝑎 ∈ ω → ∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏 ↔ 𝑏 = (2nd ‘(𝑄‘𝑎))))
53		fvex 6847	. . . . . . . 8 ⊢ (2nd ‘(𝑄‘𝑎)) ∈ V
54		eqeq2 2749	. . . . . . . . . 10 ⊢ (𝑐 = (2nd ‘(𝑄‘𝑎)) → (𝑏 = 𝑐 ↔ 𝑏 = (2nd ‘(𝑄‘𝑎))))
55	54	bibi2d 342	. . . . . . . . 9 ⊢ (𝑐 = (2nd ‘(𝑄‘𝑎)) → ((𝑎ran (𝑄 ↾ ω)𝑏 ↔ 𝑏 = 𝑐) ↔ (𝑎ran (𝑄 ↾ ω)𝑏 ↔ 𝑏 = (2nd ‘(𝑄‘𝑎)))))
56	55	albidv 1922	. . . . . . . 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 3549	. . . . . . 7 ⊢ (∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏 ↔ 𝑏 = (2nd ‘(𝑄‘𝑎))) → ∃𝑐∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏 ↔ 𝑏 = 𝑐))
58	52, 57	syl 17	. . . . . 6 ⊢ (𝑎 ∈ ω → ∃𝑐∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏 ↔ 𝑏 = 𝑐))
59		eu6 2575	. . . . . 6 ⊢ (∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏 ↔ ∃𝑐∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏 ↔ 𝑏 = 𝑐))
60	58, 59	sylibr 234	. . . . 5 ⊢ (𝑎 ∈ ω → ∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏)
61	60	rgen 3054	. . . 4 ⊢ ∀𝑎 ∈ ω ∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏
62		dff3 7046	. . . 4 ⊢ (ran (𝑄 ↾ ω):ω⟶V ↔ (ran (𝑄 ↾ ω) ⊆ (ω × V) ∧ ∀𝑎 ∈ ω ∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏))
63	17, 61, 62	mpbir2an 712	. . 3 ⊢ ran (𝑄 ↾ ω):ω⟶V
64		df-ima 5637	. . . 4 ⊢ (𝑄 “ ω) = ran (𝑄 ↾ ω)
65	64	feq1i 6653	. . 3 ⊢ ((𝑄 “ ω):ω⟶V ↔ ran (𝑄 ↾ ω):ω⟶V)
66	63, 65	mpbir 231	. 2 ⊢ (𝑄 “ ω):ω⟶V
67		dffn2 6664	. 2 ⊢ ((𝑄 “ ω) Fn ω ↔ (𝑄 “ ω):ω⟶V)
68	66, 67	mpbir 231	1 ⊢ (𝑄 “ ω) Fn ω

Syntax hints:   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃!weu 2569  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ⊆ wss 3890  ∅c0 4274  ⟨cop 4574   class class class wbr 5086   I cid 5518   × cxp 5622  ran crn 5625   ↾ cres 5626   “ cima 5627  suc csuc 6319   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  ωcom 7810  2^nd c2nd 7934  reccrdg 8341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342

This theorem is referenced by:  seqomlem3  8384  seqomlem4  8385  fnseqom  8387