Theorem rdglim2 3949

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 3948	. 2 |- ((B e. C /\ Lim B) -> (rec(F, A)` B) = U.(rec(F, A)"B))
2		limord 3028	. . . . . . . . . . 11 |- (Lim B -> Ord B)
3		ordelord 2970	. . . . . . . . . . . . 13 |- ((Ord B /\ x e. B) -> Ord x)
4	3	ex 373	. . . . . . . . . . . 12 |- (Ord B -> (x e. B -> Ord x))
5		visset 1813	. . . . . . . . . . . . 13 |- x e. V
6	5	elon 2957	. . . . . . . . . . . 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 3939	. . . . . . . . . . . 12 |- rec(F, A) Fn On
10		visset 1813	. . . . . . . . . . . . 13 |- y e. V
11	10	fnopfvb 3754	. . . . . . . . . . . 12 |- ((rec(F, A) Fn On /\ x e. On) -> ((rec(F, A)` x) = y <-> <.x, y>. e. rec(F, A)))
12	9, 11	mpan 695	. . . . . . . . . . 11 |- (x e. On -> ((rec(F, A)` x) = y <-> <.x, y>. e. rec(F, A)))
13		eqcom 1477	. . . . . . . . . . 11 |- (y = (rec(F, A)` x) <-> (rec(F, A)` x) = y)
14	12, 13	syl5bb 532	. . . . . . . . . 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 647	. . . . . . . 8 |- (Lim B -> ((x e. B /\ y = (rec(F, A)` x)) <-> (x e. B /\ <.x, y>. e. rec(F, A))))
17	16	exbidv 1279	. . . . . . 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 1650	. . . . . . 7 |- (E.x e. B y = (rec(F, A)` x) <-> E.x(x e. B /\ y = (rec(F, A)` x)))
19	17, 18	syl5rbb 533	. . . . . 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 1577	. . . . 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 3406	. . . . 5 |- (rec(F, A)"B) = {y | E.x(x e. B /\ <.x, y>. e. rec(F, A))}
22	20, 21	syl5eq 1519	. . . 4 |- (Lim B -> (rec(F, A)"B) = {y | E.x e. B y = (rec(F, A)` x)})
23	22	unieqd 2512	. . 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 1507	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 956   e. wcel 958  E.wex 980  {cab 1463  E.wrex 1646  <.cop 2411  U.cuni 2503  Ord word 2947  Oncon0 2948  Lim wlim 2949  "cima 3173   Fn wfn 3177  ` cfv 3182  reccrdg 3931

This theorem is referenced by:  rdglim2a 3950

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932

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