Theorem preddowncl 13907

Description: A property of classes that are downward closed under predecessor.

Assertion

Ref	Expression
preddowncl	|- ((B C_ A /\ A.x e. B Pred(R, A, x) C_ B) -> (X e. B -> Pred(R, B, X) = Pred(R, A, X)))

Distinct variable groups:   x,A   x,B   x,R

Proof of Theorem preddowncl

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . . 5 |- (y = X -> (y e. B <-> X e. B))
2		predeq3 13883	. . . . . 6 |- (y = X -> Pred(R, B, y) = Pred(R, B, X))
3		predeq3 13883	. . . . . 6 |- (y = X -> Pred(R, A, y) = Pred(R, A, X))
4	2, 3	eqeq12d 1899	. . . . 5 |- (y = X -> (Pred(R, B, y) = Pred(R, A, y) <-> Pred(R, B, X) = Pred(R, A, X)))
5	1, 4	imbi12d 688	. . . 4 |- (y = X -> ((y e. B -> Pred(R, B, y) = Pred(R, A, y)) <-> (X e. B -> Pred(R, B, X) = Pred(R, A, X))))
6	5	imbi2d 674	. . 3 |- (y = X -> (((B C_ A /\ A.x e. B Pred(R, A, x) C_ B) -> (y e. B -> Pred(R, B, y) = Pred(R, A, y))) <-> ((B C_ A /\ A.x e. B Pred(R, A, x) C_ B) -> (X e. B -> Pred(R, B, X) = Pred(R, A, X)))))
7		predpredss 13884	. . . . . 6 |- (B C_ A -> Pred(R, B, y) C_ Pred(R, A, y))
8	7	ad2antrr 440	. . . . 5 |- (((B C_ A /\ A.x e. B Pred(R, A, x) C_ B) /\ y e. B) -> Pred(R, B, y) C_ Pred(R, A, y))
9		predeq3 13883	. . . . . . . . . . . . 13 |- (x = y -> Pred(R, A, x) = Pred(R, A, y))
10	9	sseq1d 2644	. . . . . . . . . . . 12 |- (x = y -> (Pred(R, A, x) C_ B <-> Pred(R, A, y) C_ B))
11	10	rcla4cv 2377	. . . . . . . . . . 11 |- (A.x e. B Pred(R, A, x) C_ B -> (y e. B -> Pred(R, A, y) C_ B))
12	11	imp 377	. . . . . . . . . 10 |- ((A.x e. B Pred(R, A, x) C_ B /\ y e. B) -> Pred(R, A, y) C_ B)
13	12	sseld 2619	. . . . . . . . 9 |- ((A.x e. B Pred(R, A, x) C_ B /\ y e. B) -> (z e. Pred(R, A, y) -> z e. B))
14		visset 2295	. . . . . . . . . . 11 |- y e. _V
15		visset 2295	. . . . . . . . . . 11 |- z e. _V
16	14, 15	predbr 13896	. . . . . . . . . 10 |- (z e. Pred(R, A, y) -> zRy)
17	16	a1i 8	. . . . . . . . 9 |- ((A.x e. B Pred(R, A, x) C_ B /\ y e. B) -> (z e. Pred(R, A, y) -> zRy))
18	13, 17	jcad 661	. . . . . . . 8 |- ((A.x e. B Pred(R, A, x) C_ B /\ y e. B) -> (z e. Pred(R, A, y) -> (z e. B /\ zRy)))
19	15	elpred 13888	. . . . . . . . . 10 |- (y e. B -> (z e. Pred(R, B, y) <-> (z e. B /\ zRy)))
20	19	imbi2d 674	. . . . . . . . 9 |- (y e. B -> ((z e. Pred(R, A, y) -> z e. Pred(R, B, y)) <-> (z e. Pred(R, A, y) -> (z e. B /\ zRy))))
21	20	adantl 424	. . . . . . . 8 |- ((A.x e. B Pred(R, A, x) C_ B /\ y e. B) -> ((z e. Pred(R, A, y) -> z e. Pred(R, B, y)) <-> (z e. Pred(R, A, y) -> (z e. B /\ zRy))))
22	18, 21	mpbird 213	. . . . . . 7 |- ((A.x e. B Pred(R, A, x) C_ B /\ y e. B) -> (z e. Pred(R, A, y) -> z e. Pred(R, B, y)))
23	22	ssrdv 2622	. . . . . 6 |- ((A.x e. B Pred(R, A, x) C_ B /\ y e. B) -> Pred(R, A, y) C_ Pred(R, B, y))
24	23	adantll 428	. . . . 5 |- (((B C_ A /\ A.x e. B Pred(R, A, x) C_ B) /\ y e. B) -> Pred(R, A, y) C_ Pred(R, B, y))
25	8, 24	eqssd 2633	. . . 4 |- (((B C_ A /\ A.x e. B Pred(R, A, x) C_ B) /\ y e. B) -> Pred(R, B, y) = Pred(R, A, y))
26	25	ex 402	. . 3 |- ((B C_ A /\ A.x e. B Pred(R, A, x) C_ B) -> (y e. B -> Pred(R, B, y) = Pred(R, A, y)))
27	6, 26	vtoclg 2346	. 2 |- (X e. B -> ((B C_ A /\ A.x e. B Pred(R, A, x) C_ B) -> (X e. B -> Pred(R, B, X) = Pred(R, A, X))))
28	27	pm2.43b 81	1 |- ((B C_ A /\ A.x e. B Pred(R, A, x) C_ B) -> (X e. B -> Pred(R, B, X) = Pred(R, A, X)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593   class class class wbr 3338  Predcpred 13879

This theorem is referenced by:  wfrlem4 13960

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-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-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-pred 13880

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