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Theorem seqomlem3 8454
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 7881 . . . . . . 7 ∅ ∈ ω
2 fvres 6910 . . . . . . 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 6892 . . . . . 6 (𝑄‘∅) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅)
6 opex 5464 . . . . . . 7 ⟨∅, ( I ‘𝐼)⟩ ∈ V
76rdg0 8423 . . . . . 6 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅) = ⟨∅, ( I ‘𝐼)⟩
83, 5, 73eqtri 2764 . . . . 5 ((𝑄 ↾ ω)‘∅) = ⟨∅, ( I ‘𝐼)⟩
9 frfnom 8437 . . . . . . 7 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
104reseq1i 5977 . . . . . . . 8 (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
1110fneq1i 6646 . . . . . . 7 ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω)
129, 11mpbir 230 . . . . . 6 (𝑄 ↾ ω) Fn ω
13 fnfvelrn 7082 . . . . . 6 (((𝑄 ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω))
1412, 1, 13mp2an 690 . . . . 5 ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω)
158, 14eqeltrri 2830 . . . 4 ⟨∅, ( I ‘𝐼)⟩ ∈ ran (𝑄 ↾ ω)
16 df-ima 5689 . . . 4 (𝑄 “ ω) = ran (𝑄 ↾ ω)
1715, 16eleqtrri 2832 . . 3 ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω)
18 df-br 5149 . . 3 (∅(𝑄 “ ω)( I ‘𝐼) ↔ ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω))
1917, 18mpbir 230 . 2 ∅(𝑄 “ ω)( I ‘𝐼)
204seqomlem2 8453 . . 3 (𝑄 “ ω) Fn ω
21 fnbrfvb 6944 . . 3 (((𝑄 “ ω) Fn ω ∧ ∅ ∈ ω) → (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼)))
2220, 1, 21mp2an 690 . 2 (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼))
2319, 22mpbir 230 1 ((𝑄 “ ω)‘∅) = ( I ‘𝐼)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  wcel 2106  Vcvv 3474  c0 4322  cop 4634   class class class wbr 5148   I cid 5573  ran crn 5677  cres 5678  cima 5679  suc csuc 6366   Fn wfn 6538  cfv 6543  (class class class)co 7411  cmpo 7413  ωcom 7857  reccrdg 8411
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412
This theorem is referenced by:  seqom0g  8458
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