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Theorem seqomlem3 8345
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 7795 . . . . . . 7 ∅ ∈ ω
2 fvres 6838 . . . . . . 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 6820 . . . . . 6 (𝑄‘∅) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅)
6 opex 5403 . . . . . . 7 ⟨∅, ( I ‘𝐼)⟩ ∈ V
76rdg0 8314 . . . . . 6 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)‘∅) = ⟨∅, ( I ‘𝐼)⟩
83, 5, 73eqtri 2768 . . . . 5 ((𝑄 ↾ ω)‘∅) = ⟨∅, ( I ‘𝐼)⟩
9 frfnom 8328 . . . . . . 7 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
104reseq1i 5913 . . . . . . . 8 (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
1110fneq1i 6576 . . . . . . 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 7008 . . . . . 6 (((𝑄 ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω))
1412, 1, 13mp2an 689 . . . . 5 ((𝑄 ↾ ω)‘∅) ∈ ran (𝑄 ↾ ω)
158, 14eqeltrri 2834 . . . 4 ⟨∅, ( I ‘𝐼)⟩ ∈ ran (𝑄 ↾ ω)
16 df-ima 5627 . . . 4 (𝑄 “ ω) = ran (𝑄 ↾ ω)
1715, 16eleqtrri 2836 . . 3 ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω)
18 df-br 5090 . . 3 (∅(𝑄 “ ω)( I ‘𝐼) ↔ ⟨∅, ( I ‘𝐼)⟩ ∈ (𝑄 “ ω))
1917, 18mpbir 230 . 2 ∅(𝑄 “ ω)( I ‘𝐼)
204seqomlem2 8344 . . 3 (𝑄 “ ω) Fn ω
21 fnbrfvb 6872 . . 3 (((𝑄 “ ω) Fn ω ∧ ∅ ∈ ω) → (((𝑄 “ ω)‘∅) = ( I ‘𝐼) ↔ ∅(𝑄 “ ω)( I ‘𝐼)))
2220, 1, 21mp2an 689 . 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 1540  wcel 2105  Vcvv 3441  c0 4268  cop 4578   class class class wbr 5089   I cid 5511  ran crn 5615  cres 5616  cima 5617  suc csuc 6298   Fn wfn 6468  cfv 6473  (class class class)co 7329  cmpo 7331  ωcom 7772  reccrdg 8302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pr 5369  ax-un 7642
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-ov 7332  df-oprab 7333  df-mpo 7334  df-om 7773  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303
This theorem is referenced by:  seqom0g  8349
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