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Theorem seqomlem3 8077
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 7590 . . . . . . 7 ∅ ∈ ω
2 fvres 6682 . . . . . . 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 6664 . . . . . 6 (𝑄‘∅) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅)
6 opex 5347 . . . . . . 7 ⟨∅, ( I ‘𝐼)⟩ ∈ V
76rdg0 8046 . . . . . 6 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅) = ⟨∅, ( I ‘𝐼)⟩
83, 5, 73eqtri 2845 . . . . 5 ((𝑄 ↾ ω)‘∅) = ⟨∅, ( I ‘𝐼)⟩
9 frfnom 8059 . . . . . . 7 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
104reseq1i 5842 . . . . . . . 8 (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
1110fneq1i 6443 . . . . . . 7 ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω)
129, 11mpbir 232 . . . . . 6 (𝑄 ↾ ω) Fn ω
13 fnfvelrn 6840 . . . . . 6 (((𝑄 ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω))
1412, 1, 13mp2an 688 . . . . 5 ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω)
158, 14eqeltrri 2907 . . . 4 ⟨∅, ( I ‘𝐼)⟩ ∈ ran (𝑄 ↾ ω)
16 df-ima 5561 . . . 4 (𝑄 “ ω) = ran (𝑄 ↾ ω)
1715, 16eleqtrri 2909 . . 3 ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω)
18 df-br 5058 . . 3 (∅(𝑄 “ ω)( I ‘𝐼) ↔ ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω))
1917, 18mpbir 232 . 2 ∅(𝑄 “ ω)( I ‘𝐼)
204seqomlem2 8076 . . 3 (𝑄 “ ω) Fn ω
21 fnbrfvb 6711 . . 3 (((𝑄 “ ω) Fn ω ∧ ∅ ∈ ω) → (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼)))
2220, 1, 21mp2an 688 . 2 (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼))
2319, 22mpbir 232 1 ((𝑄 “ ω)‘∅) = ( I ‘𝐼)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wcel 2105  Vcvv 3492  c0 4288  cop 4563   class class class wbr 5057   I cid 5452  ran crn 5549  cres 5550  cima 5551  suc csuc 6186   Fn wfn 6343  cfv 6348  (class class class)co 7145  cmpo 7147  ωcom 7569  reccrdg 8034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035
This theorem is referenced by:  seqom0g  8081
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