Theorem rdgsuc 2980

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

Hypothesis

Ref	Expression
rdg.2	|- B e. On

Assertion

Ref	Expression
rdgsuc	|- (rec(F, A)` suc B) = (F` (rec(F, A)` B))

Proof of Theorem rdgsuc

Step	Hyp	Ref	Expression
1		rdglem2 2976	. 2 |- {<.w, z>. | ((w = (/) /\ z = A) \/ (-. (w = (/) \/ Lim dom w) /\ z = (F` (w` U.dom w))) \/ (Lim dom w /\ z = U.ran w))} = {<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (F` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))}
2		rdgfnon 2977	. 2 |- rec(F, A) Fn On
3		rdgval 2978	. 2 |- (g e. On -> (rec(F, A)` g) = ({<.w, z>. | ((w = (/) /\ z = A) \/ (-. (w = (/) \/ Lim dom w) /\ z = (F` (w` U.dom w))) \/ (Lim dom w /\ z = U.ran w))}` (rec(F, A) |` g)))
4		rdg.2	. 2 |- B e. On
5	1, 2, 3, 4	tz7.44-2 2967	1 |- (rec(F, A)` suc B) = (F` (rec(F, A)` B))

Syntax hints:  -. wn 1   \/ wo 195   /\ wa 196   \/ w3o 580   = wceq 1091   e. wcel 1092  (/)c0 1707  U.cuni 1919  {copab 2055  Oncon0 2199  Lim wlim 2200  suc csuc 2201  dom cdm 2410  ran crn 2411  ` cfv 2422  reccrdg 2969

This theorem is referenced by:  rdgsuct 2983

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-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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