Theorem rdgeq1 2972

Description: Equality theorem for the recursive definition generator.

Assertion

Ref	Expression
rdgeq1	|- (F = G -> rec(F, A) = rec(G, A))

Proof of Theorem rdgeq1

Step	Hyp	Ref	Expression
1		pm4.2i 149	. . . . . . . . . . 11 |- (F = G -> ((g = (/) /\ z = A) <-> (g = (/) /\ z = A)))
2		fveq1 2831	. . . . . . . . . . . . 13 |- (F = G -> (F` (g` U.dom g)) = (G` (g` U.dom g)))
3	2	cleq2d 1112	. . . . . . . . . . . 12 |- (F = G -> (z = (F` (g` U.dom g)) <-> z = (G` (g` U.dom g))))
4	3	anbi2d 468	. . . . . . . . . . 11 |- (F = G -> ((-. (g = (/) \/ Lim dom g) /\ z = (F` (g` U.dom g))) <-> (-. (g = (/) \/ Lim dom g) /\ z = (G` (g` U.dom g)))))
5		pm4.2i 149	. . . . . . . . . . 11 |- (F = G -> ((Lim dom g /\ z = U.ran g) <-> (Lim dom g /\ z = U.ran g)))
6	1, 4, 5	bi3ord 635	. . . . . . . . . 10 |- (F = G -> (((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (F` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g)) <-> ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (G` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))))
7	6	biopabdv 2102	. . . . . . . . 9 |- (F = G -> {<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (F` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))} = {<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (G` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))})
8	7	fveq1d 2834	. . . . . . . 8 |- (F = G -> ({<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (F` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))}` (f |` y)) = ({<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (G` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))}` (f |` y)))
9	8	cleq2d 1112	. . . . . . 7 |- (F = G -> ((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)) <-> (f` y) = ({<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (G` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))}` (f |` y))))
10	9	biraldv 1219	. . . . . 6 |- (F = G -> (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)) <-> A.y e. x (f` y) = ({<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (G` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))}` (f |` y))))
11	10	anbi2d 468	. . . . 5 |- (F = G -> ((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))) <-> (f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (G` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))}` (f |` y)))))
12	11	birexdv 1220	. . . 4 |- (F = G -> (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))) <-> E.x e. On (f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (G` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))}` (f |` y)))))
13	12	biabdv 1183	. . 3 |- (F = G -> {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)))} = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (G` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))}` (f |` y)))})
14	13	unieqd 1929	. 2 |- (F = G -> U.{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)))} = U.{f | E.x e. On (f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (G` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))}` (f |` y)))})
15		df-rdg 2970	. 2 |- rec(F, A) = U.{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)))}
16		df-rdg 2970	. 2 |- rec(G, A) = U.{f | E.x e. On (f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (G` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))}` (f |` y)))}
17	14, 15, 16	3eqtr4g 1147	1 |- (F = G -> rec(F, A) = rec(G, A))

Syntax hints:  -. wn 1   -> wi 2   \/ wo 195   /\ wa 196   \/ w3o 580  {cab 1090   = wceq 1091  A.wral 1201  E.wrex 1202  (/)c0 1707  U.cuni 1919  {copab 2055  Oncon0 2199  Lim wlim 2200  dom cdm 2410  ran crn 2411   |` cres 2412   Fn wfn 2417  ` cfv 2422  reccrdg 2969

This theorem is referenced by:  omv 3120  oev 3122  seqval 4665  seqsuclem 4669

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-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  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-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-rdg 2970

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