Theorem seqomlem3 8445

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 7891	. . . . . . 7 ⊢ ∅ ∈ ω
2		fvres 6904	. . . . . . 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 6886	. . . . . 6 ⊢ (𝑄‘∅) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅)
6		opex 5447	. . . . . . 7 ⊢ ⟨∅, ( I ‘𝐼)⟩ ∈ V
7	6	rdg0 8414	. . . . . 6 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅) = ⟨∅, ( I ‘𝐼)⟩
8	3, 5, 7	3eqtri 2792	. . . . 5 ⊢ ((𝑄 ↾ ω)‘∅) = ⟨∅, ( I ‘𝐼)⟩
9		frfnom 8428	. . . . . . 7 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
10	4	reseq1i 5976	. . . . . . . 8 ⊢ (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
11	10	fneq1i 6636	. . . . . . 7 ⊢ ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω)
12	9, 11	mpbir 234	. . . . . 6 ⊢ (𝑄 ↾ ω) Fn ω
13		fnfvelrn 7079	. . . . . 6 ⊢ (((𝑄 ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω))
14	12, 1, 13	mp2an 705	. . . . 5 ⊢ ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω)
15	8, 14	eqeltrri 2862	. . . 4 ⊢ ⟨∅, ( I ‘𝐼)⟩ ∈ ran (𝑄 ↾ ω)
16		df-ima 5676	. . . 4 ⊢ (𝑄 “ ω) = ran (𝑄 ↾ ω)
17	15, 16	eleqtrri 2864	. . 3 ⊢ ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω)
18		df-br 5112	. . 3 ⊢ (∅(𝑄 “ ω)( I ‘𝐼) ↔ ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω))
19	17, 18	mpbir 234	. 2 ⊢ ∅(𝑄 “ ω)( I ‘𝐼)
20	4	seqomlem2 8444	. . 3 ⊢ (𝑄 “ ω) Fn ω
21		fnbrfvb 6935	. . 3 ⊢ (((𝑄 “ ω) Fn ω ∧ ∅ ∈ ω) → (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼)))
22	20, 1, 21	mp2an 705	. 2 ⊢ (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼))
23	19, 22	mpbir 234	1 ⊢ ((𝑄 “ ω)‘∅) = ( I ‘𝐼)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ∅c0 4286  ⟨cop 4597   class class class wbr 5111   I cid 5557  ran crn 5664   ↾ cres 5665   “ cima 5666  suc csuc 6366   Fn wfn 6535  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  ωcom 7868  reccrdg 8402

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403

This theorem is used by:  seqom0g  8449