Theorem rdglim2 5157

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		rdglim 5156	. 2 |- ((B e. C /\ Lim B) -> (rec(F, A)` B) = U.(rec(F, A)"B))
2		limord 3723	. . . . . . . . . . 11 |- (Lim B -> Ord B)
3		ordelord 3680	. . . . . . . . . . . . 13 |- ((Ord B /\ x e. B) -> Ord x)
4	3	ex 402	. . . . . . . . . . . 12 |- (Ord B -> (x e. B -> Ord x))
5		visset 2295	. . . . . . . . . . . . 13 |- x e. _V
6	5	elon 3666	. . . . . . . . . . . 12 |- (x e. On <-> Ord x)
7	4, 6	syl6ibr 230	. . . . . . . . . . 11 |- (Ord B -> (x e. B -> x e. On))
8	2, 7	syl 12	. . . . . . . . . 10 |- (Lim B -> (x e. B -> x e. On))
9		rdgfnon 5147	. . . . . . . . . . . 12 |- rec(F, A) Fn On
10		visset 2295	. . . . . . . . . . . . 13 |- y e. _V
11	10	fnopfvb 4713	. . . . . . . . . . . 12 |- ((rec(F, A) Fn On /\ x e. On) -> ((rec(F, A)` x) = y <-> <.x, y>. e. rec(F, A)))
12	9, 11	mpan 759	. . . . . . . . . . 11 |- (x e. On -> ((rec(F, A)` x) = y <-> <.x, y>. e. rec(F, A)))
13		eqcom 1886	. . . . . . . . . . 11 |- (y = (rec(F, A)` x) <-> (rec(F, A)` x) = y)
14	12, 13	syl5bb 591	. . . . . . . . . 10 |- (x e. On -> (y = (rec(F, A)` x) <-> <.x, y>. e. rec(F, A)))
15	8, 14	syl6 25	. . . . . . . . 9 |- (Lim B -> (x e. B -> (y = (rec(F, A)` x) <-> <.x, y>. e. rec(F, A))))
16	15	pm5.32d 709	. . . . . . . 8 |- (Lim B -> ((x e. B /\ y = (rec(F, A)` x)) <-> (x e. B /\ <.x, y>. e. rec(F, A))))
17	16	exbidv 1657	. . . . . . 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 2110	. . . . . . 7 |- (E.x e. B y = (rec(F, A)` x) <-> E.x(x e. B /\ y = (rec(F, A)` x)))
19	17, 18	syl5rbb 592	. . . . . 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 2008	. . . . 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 4267	. . . . 5 |- (rec(F, A)"B) = {y | E.x(x e. B /\ <.x, y>. e. rec(F, A))}
22	20, 21	syl5eq 1940	. . . 4 |- (Lim B -> (rec(F, A)"B) = {y | E.x e. B y = (rec(F, A)` x)})
23	22	unieqd 3188	. . 3 |- (Lim B -> U.(rec(F, A)"B) = U.{y | E.x e. B y = (rec(F, A)` x)})
24	23	adantl 424	. 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 1925	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 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  E.wrex 2106  <.cop 3046  U.cuni 3177  Ord word 3656  Oncon0 3657  Lim wlim 3658  "cima 3989   Fn wfn 3993  ` cfv 3998  reccrdg 5139

This theorem is referenced by:  rdglim2a 5158

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140

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