Theorem seqomlem3 8463

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 7900	. . . . . . 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 5431	. . . . . . 7 ⊢ ⟨∅, ( I ‘𝐼)⟩ ∈ V
7	6	rdg0 8430	. . . . . 6 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅) = ⟨∅, ( I ‘𝐼)⟩
8	3, 5, 7	3eqtri 2787	. . . . 5 ⊢ ((𝑄 ↾ ω)‘∅) = ⟨∅, ( I ‘𝐼)⟩
9		frfnom 8444	. . . . . . 7 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
10	4	reseq1i 5965	. . . . . . . 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 7080	. . . . . 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 2857	. . . 4 ⊢ ⟨∅, ( I ‘𝐼)⟩ ∈ ran (𝑄 ↾ ω)
16		df-ima 5663	. . . 4 ⊢ (𝑄 “ ω) = ran (𝑄 ↾ ω)
17	15, 16	eleqtrri 2859	. . 3 ⊢ ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω)
18		df-br 5103	. . 3 ⊢ (∅(𝑄 “ ω)( I ‘𝐼) ↔ ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω))
19	17, 18	mpbir 234	. 2 ⊢ ∅(𝑄 “ ω)( I ‘𝐼)
20	4	seqomlem2 8462	. . 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 1569   ∈ wcel 2144  Vcvv 3450  ∅c0 4278  ⟨cop 4589   class class class wbr 5102   I cid 5544  ran crn 5651   ↾ cres 5652   “ cima 5653  suc csuc 6364   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  ωcom 7877  reccrdg 8418

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419

This theorem is used by:  seqom0g  8467