Theorem seqomlem3 8381

Description: Lemma for seq_ω. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Hypothesis

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

Assertion

Ref	Expression
seqomlem3	⊢ ((𝑄 “ ω)‘∅) = ( I ‘𝐼)

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

Allowed substitution hints:   𝐼(𝑣,𝑖)

Proof of Theorem seqomlem3

Step	Hyp	Ref	Expression
1		peano1 7829	. . . . . . 7 ⊢ ∅ ∈ ω
2		fvres 6851	. . . . . . 7 ⊢ (∅ ∈ ω → ((𝑄 ↾ ω)‘∅) = (𝑄‘∅))
3	1, 2	ax-mp 5	. . . . . 6 ⊢ ((𝑄 ↾ ω)‘∅) = (𝑄‘∅)
4		seqomlem.a	. . . . . . 7 ⊢ 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)
5	4	fveq1i 6833	. . . . . 6 ⊢ (𝑄‘∅) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅)
6		opex 5410	. . . . . . 7 ⊢ ⟨∅, ( I ‘𝐼)⟩ ∈ V
7	6	rdg0 8350	. . . . . 6 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅) = ⟨∅, ( I ‘𝐼)⟩
8	3, 5, 7	3eqtri 2761	. . . . 5 ⊢ ((𝑄 ↾ ω)‘∅) = ⟨∅, ( I ‘𝐼)⟩
9		frfnom 8364	. . . . . . 7 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
10	4	reseq1i 5932	. . . . . . . 8 ⊢ (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
11	10	fneq1i 6587	. . . . . . 7 ⊢ ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω)
12	9, 11	mpbir 231	. . . . . 6 ⊢ (𝑄 ↾ ω) Fn ω
13		fnfvelrn 7023	. . . . . 6 ⊢ (((𝑄 ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω))
14	12, 1, 13	mp2an 692	. . . . 5 ⊢ ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω)
15	8, 14	eqeltrri 2831	. . . 4 ⊢ ⟨∅, ( I ‘𝐼)⟩ ∈ ran (𝑄 ↾ ω)
16		df-ima 5635	. . . 4 ⊢ (𝑄 “ ω) = ran (𝑄 ↾ ω)
17	15, 16	eleqtrri 2833	. . 3 ⊢ ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω)
18		df-br 5097	. . 3 ⊢ (∅(𝑄 “ ω)( I ‘𝐼) ↔ ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω))
19	17, 18	mpbir 231	. 2 ⊢ ∅(𝑄 “ ω)( I ‘𝐼)
20	4	seqomlem2 8380	. . 3 ⊢ (𝑄 “ ω) Fn ω
21		fnbrfvb 6882	. . 3 ⊢ (((𝑄 “ ω) Fn ω ∧ ∅ ∈ ω) → (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼)))
22	20, 1, 21	mp2an 692	. 2 ⊢ (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼))
23	19, 22	mpbir 231	1 ⊢ ((𝑄 “ ω)‘∅) = ( I ‘𝐼)

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2113  Vcvv 3438  ∅c0 4283  ⟨cop 4584   class class class wbr 5096   I cid 5516  ran crn 5623   ↾ cres 5624   “ cima 5625  suc csuc 6317   Fn wfn 6485  ‘cfv 6490  (class class class)co 7356   ∈ cmpo 7358  ωcom 7806  reccrdg 8338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339

This theorem is referenced by:  seqom0g  8385