MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seqom0g Structured version   Visualization version   GIF version

Theorem seqom0g 8454
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 8446 . . . . 5 seqω(𝐹, 𝐼) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
31, 2eqtri 2754 . . . 4 𝐺 = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
43fveq1i 6885 . . 3 (𝐺‘∅) = ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)‘∅)
5 seqomlem0 8447 . . . 4 rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ ⟨suc 𝑐, (𝑐𝐹𝑑)⟩), ⟨∅, ( I ‘𝐼)⟩)
65seqomlem3 8450 . . 3 ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)‘∅) = ( I ‘𝐼)
74, 6eqtri 2754 . 2 (𝐺‘∅) = ( I ‘𝐼)
8 fvi 6960 . 2 (𝐼𝑉 → ( I ‘𝐼) = 𝐼)
97, 8eqtrid 2778 1 (𝐼𝑉 → (𝐺‘∅) = 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3468  c0 4317  cop 4629   I cid 5566  cima 5672  suc csuc 6359  cfv 6536  (class class class)co 7404  cmpo 7406  ωcom 7851  reccrdg 8407  seqωcseqom 8445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-seqom 8446
This theorem is referenced by:  cantnfvalf  9659  cantnfval2  9663  cantnflt  9666  cantnff  9668  cantnf0  9669  cantnfp1lem3  9674  cantnf  9687  cnfcom  9694  fseqenlem1  10018  fin23lem14  10327  fin23lem16  10329
  Copyright terms: Public domain W3C validator