Theorem seqomsuc 8484

Description: Value of an index-aware recursive definition at a successor. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Hypothesis

Ref	Expression
seqom.a	⊢ 𝐺 = seqω(𝐹, 𝐼)

Assertion

Ref	Expression
seqomsuc	⊢ (𝐴 ∈ ω → (𝐺‘suc 𝐴) = (𝐴𝐹(𝐺‘𝐴)))

Proof of Theorem seqomsuc

Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqomlem0 8476	. . 3 ⊢ rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) = rec((𝑐 ∈ ω, 𝑑 ∈ V ↦ ⟨suc 𝑐, (𝑐𝐹𝑑)⟩), ⟨∅, ( I ‘𝐼)⟩)
2	1	seqomlem4 8480	. 2 ⊢ (𝐴 ∈ ω → ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)‘suc 𝐴) = (𝐴𝐹((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)‘𝐴)))
3		seqom.a	. . . 4 ⊢ 𝐺 = seqω(𝐹, 𝐼)
4		df-seqom 8475	. . . 4 ⊢ seqω(𝐹, 𝐼) = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
5	3, 4	eqtri 2756	. . 3 ⊢ 𝐺 = (rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)
6	5	fveq1i 6903	. 2 ⊢ (𝐺‘suc 𝐴) = ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)‘suc 𝐴)
7	5	fveq1i 6903	. . 3 ⊢ (𝐺‘𝐴) = ((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)‘𝐴)
8	7	oveq2i 7437	. 2 ⊢ (𝐴𝐹(𝐺‘𝐴)) = (𝐴𝐹((rec((𝑎 ∈ ω, 𝑏 ∈ V ↦ ⟨suc 𝑎, (𝑎𝐹𝑏)⟩), ⟨∅, ( I ‘𝐼)⟩) “ ω)‘𝐴))
9	2, 6, 8	3eqtr4g 2793	1 ⊢ (𝐴 ∈ ω → (𝐺‘suc 𝐴) = (𝐴𝐹(𝐺‘𝐴)))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3473  ∅c0 4326  ⟨cop 4638   I cid 5579   “ cima 5685  suc csuc 6376  ‘cfv 6553  (class class class)co 7426   ∈ cmpo 7428  ωcom 7876  reccrdg 8436  seq_ωcseqom 8474

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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-seqom 8475

This theorem is referenced by:  cantnfvalf  9696  cantnfval2  9700  cantnfsuc  9701  cnfcomlem  9730  fseqenlem1  10055  fin23lem12  10362