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Theorem seqom0g 7759
Description: Value of an index-aware recursive definition at 0. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised 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 7751 . . . . 5 seq𝜔(𝐹, 𝐼) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
31, 2eqtri 2787 . . . 4 𝐺 = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
43fveq1i 6380 . . 3 (𝐺‘∅) = ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)‘∅)
5 seqomlem0 7752 . . . 4 rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ ⟨suc 𝑐, (𝑐𝐹𝑑)⟩), ⟨∅, ( I ‘𝐼)⟩)
65seqomlem3 7755 . . 3 ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)‘∅) = ( I ‘𝐼)
74, 6eqtri 2787 . 2 (𝐺‘∅) = ( I ‘𝐼)
8 fvi 6448 . 2 (𝐼𝑉 → ( I ‘𝐼) = 𝐼)
97, 8syl5eq 2811 1 (𝐼𝑉 → (𝐺‘∅) = 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  wcel 2155  Vcvv 3350  c0 4081  cop 4342   I cid 5186  cima 5282  suc csuc 5912  cfv 6070  (class class class)co 6846  cmpt2 6848  ωcom 7267  reccrdg 7713  seq𝜔cseqom 7750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-seqom 7751
This theorem is referenced by:  cantnfvalf  8781  cantnfval2  8785  cantnflt  8788  cantnff  8790  cantnf0  8791  cantnfp1lem3  8796  cantnf  8809  cnfcom  8816  fseqenlem1  9102  fin23lem14  9412  fin23lem16  9414
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