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Theorem seqomlem3 7434
 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 6977 . . . . . . 7 ∅ ∈ ω
2 fvres 6117 . . . . . . 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 6104 . . . . . 6 (𝑄‘∅) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅)
6 opex 4859 . . . . . . 7 ⟨∅, ( I ‘𝐼)⟩ ∈ V
76rdg0 7404 . . . . . 6 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅) = ⟨∅, ( I ‘𝐼)⟩
83, 5, 73eqtri 2636 . . . . 5 ((𝑄 ↾ ω)‘∅) = ⟨∅, ( I ‘𝐼)⟩
9 frfnom 7417 . . . . . . 7 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
104reseq1i 5313 . . . . . . . 8 (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
1110fneq1i 5899 . . . . . . 7 ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω)
129, 11mpbir 220 . . . . . 6 (𝑄 ↾ ω) Fn ω
13 fnfvelrn 6264 . . . . . 6 (((𝑄 ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω))
1412, 1, 13mp2an 704 . . . . 5 ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω)
158, 14eqeltrri 2685 . . . 4 ⟨∅, ( I ‘𝐼)⟩ ∈ ran (𝑄 ↾ ω)
16 df-ima 5051 . . . 4 (𝑄 “ ω) = ran (𝑄 ↾ ω)
1715, 16eleqtrri 2687 . . 3 ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω)
18 df-br 4584 . . 3 (∅(𝑄 “ ω)( I ‘𝐼) ↔ ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω))
1917, 18mpbir 220 . 2 ∅(𝑄 “ ω)( I ‘𝐼)
204seqomlem2 7433 . . 3 (𝑄 “ ω) Fn ω
21 fnbrfvb 6146 . . 3 (((𝑄 “ ω) Fn ω ∧ ∅ ∈ ω) → (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼)))
2220, 1, 21mp2an 704 . 2 (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼))
2319, 22mpbir 220 1 ((𝑄 “ ω)‘∅) = ( I ‘𝐼)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  ⟨cop 4131   class class class wbr 4583   I cid 4948  ran crn 5039   ↾ cres 5040   “ cima 5041  suc csuc 5642   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957  reccrdg 7392 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393 This theorem is referenced by:  seqom0g  7438
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