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Theorem seqom0g 7704
 Description: Value of an index-aware recursive definition at 0. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revise by AV, 17-Sep-2021.)
Hypothesis
Ref Expression
seqom.a 𝐺 = seq𝜔(𝐹, 𝐼)
Assertion
Ref Expression
seqom0g (𝐼𝑉 → (𝐺‘∅) = 𝐼)

Proof of Theorem seqom0g
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 seqom.a . . . . 5 𝐺 = seq𝜔(𝐹, 𝐼)
2 df-seqom 7696 . . . . 5 seq𝜔(𝐹, 𝐼) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
31, 2eqtri 2793 . . . 4 𝐺 = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
43fveq1i 6333 . . 3 (𝐺‘∅) = ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)‘∅)
5 seqomlem0 7697 . . . 4 rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ ⟨suc 𝑐, (𝑐𝐹𝑑)⟩), ⟨∅, ( I ‘𝐼)⟩)
65seqomlem3 7700 . . 3 ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)‘∅) = ( I ‘𝐼)
74, 6eqtri 2793 . 2 (𝐺‘∅) = ( I ‘𝐼)
8 fvi 6397 . 2 (𝐼𝑉 → ( I ‘𝐼) = 𝐼)
97, 8syl5eq 2817 1 (𝐼𝑉 → (𝐺‘∅) = 𝐼)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1631   ∈ wcel 2145  Vcvv 3351  ∅c0 4063  ⟨cop 4322   I cid 5156   “ cima 5252  suc csuc 5868  ‘cfv 6031  (class class class)co 6793   ↦ cmpt2 6795  ωcom 7212  reccrdg 7658  seq𝜔cseqom 7695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-seqom 7696 This theorem is referenced by:  cantnfvalf  8726  cantnfval2  8730  cantnflt  8733  cantnff  8735  cantnf0  8736  cantnfp1lem3  8741  cantnf  8754  cnfcom  8761  fseqenlem1  9047  fin23lem14  9357  fin23lem16  9359
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