Theorem wfrlem9 13965

Description: Lemma for well-founded recursion. If X e. dom F, then its predecessor class is a subset of dom F.

Hypotheses

Ref	Expression
wfrlem6.1	|- B = {f | E.x(f Fn x /\ (x C_ A /\ A.y e. x Pred(R, A, y) C_ x) /\ A.y e. x (f` y) = (G` (f |` Pred(R, A, y))))}
wfrlem6.2	|- F = U.B

Assertion

Ref	Expression
wfrlem9	|- (X e. dom F -> Pred(R, A, X) C_ dom F)

Distinct variable groups:   A,f,x,y   f,G,x,y   R,f,x,y

Proof of Theorem wfrlem9

Step	Hyp	Ref	Expression
1		wfrlem6.1	. . . . . . 7 |- B = {f | E.x(f Fn x /\ (x C_ A /\ A.y e. x Pred(R, A, y) C_ x) /\ A.y e. x (f` y) = (G` (f |` Pred(R, A, y))))}
2	1	wfrlem1 13957	. . . . . 6 |- B = {g | E.z(g Fn z /\ (z C_ A /\ A.w e. z Pred(R, A, w) C_ z) /\ A.w e. z (g` w) = (G` (g |` Pred(R, A, w))))}
3	2	abeq2i 2001	. . . . 5 |- (g e. B <-> E.z(g Fn z /\ (z C_ A /\ A.w e. z Pred(R, A, w) C_ z) /\ A.w e. z (g` w) = (G` (g |` Pred(R, A, w)))))
4		predeq3 13883	. . . . . . . . . . . 12 |- (w = X -> Pred(R, A, w) = Pred(R, A, X))
5	4	sseq1d 2644	. . . . . . . . . . 11 |- (w = X -> (Pred(R, A, w) C_ z <-> Pred(R, A, X) C_ z))
6	5	rcla4cv 2377	. . . . . . . . . 10 |- (A.w e. z Pred(R, A, w) C_ z -> (X e. z -> Pred(R, A, X) C_ z))
7	6	adantl 424	. . . . . . . . 9 |- ((g Fn z /\ A.w e. z Pred(R, A, w) C_ z) -> (X e. z -> Pred(R, A, X) C_ z))
8		fndm 4512	. . . . . . . . . . . 12 |- (g Fn z -> dom g = z)
9	8	eleq2d 1964	. . . . . . . . . . 11 |- (g Fn z -> (X e. dom g <-> X e. z))
10	8	sseq2d 2645	. . . . . . . . . . 11 |- (g Fn z -> (Pred(R, A, X) C_ dom g <-> Pred(R, A, X) C_ z))
11	9, 10	imbi12d 688	. . . . . . . . . 10 |- (g Fn z -> ((X e. dom g -> Pred(R, A, X) C_ dom g) <-> (X e. z -> Pred(R, A, X) C_ z)))
12	11	adantr 425	. . . . . . . . 9 |- ((g Fn z /\ A.w e. z Pred(R, A, w) C_ z) -> ((X e. dom g -> Pred(R, A, X) C_ dom g) <-> (X e. z -> Pred(R, A, X) C_ z)))
13	7, 12	mpbird 213	. . . . . . . 8 |- ((g Fn z /\ A.w e. z Pred(R, A, w) C_ z) -> (X e. dom g -> Pred(R, A, X) C_ dom g))
14	13	adantrl 430	. . . . . . 7 |- ((g Fn z /\ (z C_ A /\ A.w e. z Pred(R, A, w) C_ z)) -> (X e. dom g -> Pred(R, A, X) C_ dom g))
15	14	3adant3 896	. . . . . 6 |- ((g Fn z /\ (z C_ A /\ A.w e. z Pred(R, A, w) C_ z) /\ A.w e. z (g` w) = (G` (g |` Pred(R, A, w)))) -> (X e. dom g -> Pred(R, A, X) C_ dom g))
16	15	19.23aiv 1674	. . . . 5 |- (E.z(g Fn z /\ (z C_ A /\ A.w e. z Pred(R, A, w) C_ z) /\ A.w e. z (g` w) = (G` (g |` Pred(R, A, w)))) -> (X e. dom g -> Pred(R, A, X) C_ dom g))
17	3, 16	sylbi 216	. . . 4 |- (g e. B -> (X e. dom g -> Pred(R, A, X) C_ dom g))
18	17	reximia 2196	. . 3 |- (E.g e. B X e. dom g -> E.g e. B Pred(R, A, X) C_ dom g)
19		ssiun 3293	. . 3 |- (E.g e. B Pred(R, A, X) C_ dom g -> Pred(R, A, X) C_ U_g e. B dom g)
20	18, 19	syl 12	. 2 |- (E.g e. B X e. dom g -> Pred(R, A, X) C_ U_g e. B dom g)
21		wfrlem6.2	. . . . . 6 |- F = U.B
22	21	dmeqi 4158	. . . . 5 |- dom F = dom U. B
23		dmuni 4166	. . . . 5 |- dom U. B = U_g e. B dom g
24	22, 23	eqtri 1908	. . . 4 |- dom F = U_g e. B dom g
25	24	eleq2i 1961	. . 3 |- (X e. dom F <-> X e. U_g e. B dom g)
26		eliun 3259	. . 3 |- (X e. U_g e. B dom g <-> E.g e. B X e. dom g)
27	25, 26	bitri 190	. 2 |- (X e. dom F <-> E.g e. B X e. dom g)
28	24	sseq2i 2642	. 2 |- (Pred(R, A, X) C_ dom F <-> Pred(R, A, X) C_ U_g e. B dom g)
29	20, 27, 28	3imtr4i 236	1 |- (X e. dom F -> Pred(R, A, X) C_ dom F)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  A.wral 2105  E.wrex 2106   C_ wss 2593  U.cuni 3177  U_ciun 3255  dom cdm 3986   |` cres 3988   Fn wfn 3993  ` cfv 3998  Predcpred 13879

This theorem is referenced by:  wfrlem10 13966  wfrlem14 13970  wfrlem15 13971

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-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-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-id 3586  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-pred 13880

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