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Theorem seqomlem3 8075
Description: Lemma for seqω. (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
StepHypRef Expression
1 peano1 7586 . . . . . . 7 ∅ ∈ ω
2 fvres 6671 . . . . . . 7 (∅ ∈ ω → ((𝑄 ↾ ω)‘∅) = (𝑄‘∅))
31, 2ax-mp 5 . . . . . 6 ((𝑄 ↾ ω)‘∅) = (𝑄‘∅)
4 seqomlem.a . . . . . . 7 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)
54fveq1i 6653 . . . . . 6 (𝑄‘∅) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅)
6 opex 5333 . . . . . . 7 ⟨∅, ( I ‘𝐼)⟩ ∈ V
76rdg0 8044 . . . . . 6 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅) = ⟨∅, ( I ‘𝐼)⟩
83, 5, 73eqtri 2849 . . . . 5 ((𝑄 ↾ ω)‘∅) = ⟨∅, ( I ‘𝐼)⟩
9 frfnom 8057 . . . . . . 7 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
104reseq1i 5827 . . . . . . . 8 (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
1110fneq1i 6429 . . . . . . 7 ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω)
129, 11mpbir 234 . . . . . 6 (𝑄 ↾ ω) Fn ω
13 fnfvelrn 6830 . . . . . 6 (((𝑄 ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω))
1412, 1, 13mp2an 691 . . . . 5 ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω)
158, 14eqeltrri 2911 . . . 4 ⟨∅, ( I ‘𝐼)⟩ ∈ ran (𝑄 ↾ ω)
16 df-ima 5545 . . . 4 (𝑄 “ ω) = ran (𝑄 ↾ ω)
1715, 16eleqtrri 2913 . . 3 ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω)
18 df-br 5043 . . 3 (∅(𝑄 “ ω)( I ‘𝐼) ↔ ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω))
1917, 18mpbir 234 . 2 ∅(𝑄 “ ω)( I ‘𝐼)
204seqomlem2 8074 . . 3 (𝑄 “ ω) Fn ω
21 fnbrfvb 6700 . . 3 (((𝑄 “ ω) Fn ω ∧ ∅ ∈ ω) → (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼)))
2220, 1, 21mp2an 691 . 2 (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼))
2319, 22mpbir 234 1 ((𝑄 “ ω)‘∅) = ( I ‘𝐼)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wcel 2114  Vcvv 3469  c0 4265  cop 4545   class class class wbr 5042   I cid 5436  ran crn 5533  cres 5534  cima 5535  suc csuc 6171   Fn wfn 6329  cfv 6334  (class class class)co 7140  cmpo 7142  ωcom 7565  reccrdg 8032
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033
This theorem is referenced by:  seqom0g  8079
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