Theorem rdglim2 3934

Description: The value of the recursive definition generator at a limit ordinal, in terms of the union of all smaller values.

Assertion

Ref	Expression
rdglim2	|- ((B e. C /\ Lim B) -> (rec(F, A)` B) = U.{y | E.x e. B y = (rec(F, A)` x)})

Distinct variable groups:   x,y,A   x,B,y   x,F,y

Proof of Theorem rdglim2

Step	Hyp	Ref	Expression
1		rdglimt 3933	. 2 |- ((B e. C /\ Lim B) -> (rec(F, A)` B) = U.(rec(F, A)"B))
2		limord 3018	. . . . . . . . . . 11 |- (Lim B -> Ord B)
3		ordelord 2960	. . . . . . . . . . . . 13 |- ((Ord B /\ x e. B) -> Ord x)
4	3	ex 373	. . . . . . . . . . . 12 |- (Ord B -> (x e. B -> Ord x))
5		visset 1804	. . . . . . . . . . . . 13 |- x e. V
6	5	elon 2947	. . . . . . . . . . . 12 |- (x e. On <-> Ord x)
7	4, 6	syl6ibr 213	. . . . . . . . . . 11 |- (Ord B -> (x e. B -> x e. On))
8	2, 7	syl 10	. . . . . . . . . 10 |- (Lim B -> (x e. B -> x e. On))
9		rdgfnon 3924	. . . . . . . . . . . 12 |- rec(F, A) Fn On
10		visset 1804	. . . . . . . . . . . . 13 |- y e. V
11	10	fnopfvb 3739	. . . . . . . . . . . 12 |- ((rec(F, A) Fn On /\ x e. On) -> ((rec(F, A)` x) = y <-> <.x, y>. e. rec(F, A)))
12	9, 11	mpan 693	. . . . . . . . . . 11 |- (x e. On -> ((rec(F, A)` x) = y <-> <.x, y>. e. rec(F, A)))
13		eqcom 1469	. . . . . . . . . . 11 |- (y = (rec(F, A)` x) <-> (rec(F, A)` x) = y)
14	12, 13	syl5bb 530	. . . . . . . . . 10 |- (x e. On -> (y = (rec(F, A)` x) <-> <.x, y>. e. rec(F, A)))
15	8, 14	syl6 22	. . . . . . . . 9 |- (Lim B -> (x e. B -> (y = (rec(F, A)` x) <-> <.x, y>. e. rec(F, A))))
16	15	pm5.32d 645	. . . . . . . 8 |- (Lim B -> ((x e. B /\ y = (rec(F, A)` x)) <-> (x e. B /\ <.x, y>. e. rec(F, A))))
17	16	exbidv 1274	. . . . . . 7 |- (Lim B -> (E.x(x e. B /\ y = (rec(F, A)` x)) <-> E.x(x e. B /\ <.x, y>. e. rec(F, A))))
18		df-rex 1642	. . . . . . 7 |- (E.x e. B y = (rec(F, A)` x) <-> E.x(x e. B /\ y = (rec(F, A)` x)))
19	17, 18	syl5rbb 531	. . . . . 6 |- (Lim B -> (E.x(x e. B /\ <.x, y>. e. rec(F, A)) <-> E.x e. B y = (rec(F, A)` x)))
20	19	abbidv 1569	. . . . 5 |- (Lim B -> {y | E.x(x e. B /\ <.x, y>. e. rec(F, A))} = {y | E.x e. B y = (rec(F, A)` x)})
21		dfima3 3390	. . . . 5 |- (rec(F, A)"B) = {y | E.x(x e. B /\ <.x, y>. e. rec(F, A))}
22	20, 21	syl5eq 1511	. . . 4 |- (Lim B -> (rec(F, A)"B) = {y | E.x e. B y = (rec(F, A)` x)})
23	22	unieqd 2502	. . 3 |- (Lim B -> U.(rec(F, A)"B) = U.{y | E.x e. B y = (rec(F, A)` x)})
24	23	adantl 388	. 2 |- ((B e. C /\ Lim B) -> U.(rec(F, A)"B) = U.{y | E.x e. B y = (rec(F, A)` x)})
25	1, 24	eqtrd 1499	1 |- ((B e. C /\ Lim B) -> (rec(F, A)` B) = U.{y | E.x e. B y = (rec(F, A)` x)})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  {cab 1456  E.wrex 1638  <.cop 2401  U.cuni 2493  Ord word 2937  Oncon0 2938  Lim wlim 2939  "cima 3163   Fn wfn 3167  ` cfv 3172  reccrdg 3916

This theorem is referenced by:  rdglim2a 3935

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917

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