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Theorem seqom0g 7943
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 7935 . . . . 5 seq𝜔(𝐹, 𝐼) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
31, 2eqtri 2819 . . . 4 𝐺 = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
43fveq1i 6539 . . 3 (𝐺‘∅) = ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)‘∅)
5 seqomlem0 7936 . . . 4 rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ ⟨suc 𝑐, (𝑐𝐹𝑑)⟩), ⟨∅, ( I ‘𝐼)⟩)
65seqomlem3 7939 . . 3 ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)‘∅) = ( I ‘𝐼)
74, 6eqtri 2819 . 2 (𝐺‘∅) = ( I ‘𝐼)
8 fvi 6607 . 2 (𝐼𝑉 → ( I ‘𝐼) = 𝐼)
97, 8syl5eq 2843 1 (𝐼𝑉 → (𝐺‘∅) = 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1522  wcel 2081  Vcvv 3437  c0 4211  cop 4478   I cid 5347  cima 5446  suc csuc 6068  cfv 6225  (class class class)co 7016  cmpo 7018  ωcom 7436  reccrdg 7897  seq𝜔cseqom 7934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-seqom 7935
This theorem is referenced by:  cantnfvalf  8974  cantnfval2  8978  cantnflt  8981  cantnff  8983  cantnf0  8984  cantnfp1lem3  8989  cantnf  9002  cnfcom  9009  fseqenlem1  9296  fin23lem14  9601  fin23lem16  9603
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