Theorem rdgsuct 2983

Description: The value of the recursive definition generator at a successor.

Assertion

Ref	Expression
rdgsuct	⊢ (B ∈ On → (rec(F, A) ‘suc B) = (F ‘(rec(F, A) ‘B)))

Proof of Theorem rdgsuct

Step	Hyp	Ref	Expression
1		suceq 2288	. . . 4 ⊢ (B = if(B ∈ On, B, ∅) → suc B = suc if(B ∈ On, B, ∅))
2	1	fveq2d 2836	. . 3 ⊢ (B = if(B ∈ On, B, ∅) → (rec(F, A) ‘suc B) = (rec(F, A) ‘suc if(B ∈ On, B, ∅)))
3		fveq2 2832	. . . 4 ⊢ (B = if(B ∈ On, B, ∅) → (rec(F, A) ‘B) = (rec(F, A) ‘if(B ∈ On, B, ∅)))
4	3	fveq2d 2836	. . 3 ⊢ (B = if(B ∈ On, B, ∅) → (F ‘(rec(F, A) ‘B)) = (F ‘(rec(F, A) ‘if(B ∈ On, B, ∅))))
5	2, 4	cleq12d 1115	. 2 ⊢ (B = if(B ∈ On, B, ∅) → ((rec(F, A) ‘suc B) = (F ‘(rec(F, A) ‘B)) ↔ (rec(F, A) ‘suc if(B ∈ On, B, ∅)) = (F ‘(rec(F, A) ‘if(B ∈ On, B, ∅)))))
6		0elon 2277	. . . 4 ⊢ ∅ ∈ On
7	6	elimel 1793	. . 3 ⊢ if(B ∈ On, B, ∅) ∈ On
8	7	rdgsuc 2980	. 2 ⊢ (rec(F, A) ‘suc if(B ∈ On, B, ∅)) = (F ‘(rec(F, A) ‘if(B ∈ On, B, ∅)))
9	5, 8	dedth 1784	1 ⊢ (B ∈ On → (rec(F, A) ‘suc B) = (F ‘(rec(F, A) ‘B)))

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  ∅c0 1707  ifcif 1776  Oncon0 2199  suc csuc 2201   ‘cfv 2422  reccrdg 2969

This theorem is referenced by:  rdgsucopab 2984  rdgsucopabn 2985  frsuc 2991  oasuc 3131  omsuc 3133  oesuc 3134  uzrdgsuc 4659

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970

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