Theorem rdgval 3946

Description: Value of the recursive definition generator.

Assertion

Ref	Expression
rdgval	|- (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)))

Distinct variable groups:   z,g,w,F   z,A,g,w

Proof of Theorem rdgval

Step	Hyp	Ref	Expression
1		rdglem1 3943	. . 3 |- {w | E.u e. On (w Fn u /\ A.v e. u (w` v) = ({<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (F` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))}` (w |` v)))} = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (F` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))}` (f |` y)))}
2		dfrdg2 3939	. . 3 |- rec(F, A) = U.{w | E.u e. On (w Fn u /\ A.v e. u (w` v) = ({<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (F` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))}` (w |` v)))}
3	1, 2	tfr2 3931	. 2 |- (g e. On -> (rec(F, A)` g) = ({<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (F` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))}` (rec(F, A) |` g)))
4		rdglem2 3944	. . 3 |- {<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (F` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))} = {<.w, z>. | ((w = (/) /\ z = A) \/ (-. (w = (/) \/ Lim dom w) /\ z = (F` (w` U.dom w))) \/ (Lim dom w /\ z = U.ran w))}
5	4	fveq1i 3731	. 2 |- ({<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (F` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))}` (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))
6	3, 5	syl6eq 1526	1 |- (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)))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   \/ w3o 776   = wceq 958   e. wcel 960  {cab 1466  A.wral 1648  E.wrex 1649  (/)c0 2283  U.cuni 2507  {copab 2671  Oncon0 2954  Lim wlim 2955  dom cdm 3176  ran crn 3177   |` cres 3178   Fn wfn 3183  ` cfv 3188  reccrdg 3937

This theorem is referenced by:  rdg0 3947  rdgsuc 3948  rdglim 3949

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938

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