Theorem seqomlem3 8071

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 7581	. . . . . . 7 ⊢ ∅ ∈ ω
2		fvres 6664	. . . . . . 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 6646	. . . . . 6 ⊢ (𝑄‘∅) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅)
6		opex 5321	. . . . . . 7 ⊢ ⟨∅, ( I ‘𝐼)⟩ ∈ V
7	6	rdg0 8040	. . . . . 6 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅) = ⟨∅, ( I ‘𝐼)⟩
8	3, 5, 7	3eqtri 2825	. . . . 5 ⊢ ((𝑄 ↾ ω)‘∅) = ⟨∅, ( I ‘𝐼)⟩
9		frfnom 8053	. . . . . . 7 ⊢ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
10	4	reseq1i 5814	. . . . . . . 8 ⊢ (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
11	10	fneq1i 6420	. . . . . . 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 6825	. . . . . 6 ⊢ (((𝑄 ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω))
14	12, 1, 13	mp2an 691	. . . . 5 ⊢ ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω)
15	8, 14	eqeltrri 2887	. . . 4 ⊢ ⟨∅, ( I ‘𝐼)⟩ ∈ ran (𝑄 ↾ ω)
16		df-ima 5532	. . . 4 ⊢ (𝑄 “ ω) = ran (𝑄 ↾ ω)
17	15, 16	eleqtrri 2889	. . 3 ⊢ ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω)
18		df-br 5031	. . 3 ⊢ (∅(𝑄 “ ω)( I ‘𝐼) ↔ ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω))
19	17, 18	mpbir 234	. 2 ⊢ ∅(𝑄 “ ω)( I ‘𝐼)
20	4	seqomlem2 8070	. . 3 ⊢ (𝑄 “ ω) Fn ω
21		fnbrfvb 6693	. . 3 ⊢ (((𝑄 “ ω) Fn ω ∧ ∅ ∈ ω) → (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼)))
22	20, 1, 21	mp2an 691	. 2 ⊢ (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼))
23	19, 22	mpbir 234	1 ⊢ ((𝑄 “ ω)‘∅) = ( I ‘𝐼)

Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ∅c0 4243  ⟨cop 4531   class class class wbr 5030   I cid 5424  ran crn 5520   ↾ cres 5521   “ cima 5522  suc csuc 6161   Fn wfn 6319  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  ωcom 7560  reccrdg 8028

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029

This theorem is referenced by:  seqom0g  8075