Theorem setlikess 13906

Description: If R is set-like over A, then it is set-like over any subclass of A.

Assertion

Ref	Expression
setlikess	|- ((A C_ B /\ A.x e. B Pred(R, B, x) e. _V) -> A.x e. A Pred(R, A, x) e. _V)

Distinct variable groups:   x,A   x,B

Proof of Theorem setlikess

Step	Hyp	Ref	Expression
1		ssralv 2672	. . 3 |- (A C_ B -> (A.x e. B Pred(R, B, x) e. _V -> A.x e. A Pred(R, B, x) e. _V))
2		predpredss 13884	. . . . 5 |- (A C_ B -> Pred(R, A, x) C_ Pred(R, B, x))
3		ssexg 3457	. . . . . 6 |- ((Pred(R, A, x) C_ Pred(R, B, x) /\ Pred(R, B, x) e. _V) -> Pred(R, A, x) e. _V)
4	3	ex 402	. . . . 5 |- (Pred(R, A, x) C_ Pred(R, B, x) -> (Pred(R, B, x) e. _V -> Pred(R, A, x) e. _V))
5	2, 4	syl 12	. . . 4 |- (A C_ B -> (Pred(R, B, x) e. _V -> Pred(R, A, x) e. _V))
6	5	ralimdv 2172	. . 3 |- (A C_ B -> (A.x e. A Pred(R, B, x) e. _V -> A.x e. A Pred(R, A, x) e. _V))
7	1, 6	syld 30	. 2 |- (A C_ B -> (A.x e. B Pred(R, B, x) e. _V -> A.x e. A Pred(R, A, x) e. _V))
8	7	imp 377	1 |- ((A C_ B /\ A.x e. B Pred(R, B, x) e. _V) -> A.x e. A Pred(R, A, x) e. _V)

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593  Predcpred 13879

This theorem is referenced by:  wfrlem5 13961

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-v 2294  df-in 2603  df-ss 2605  df-pred 13880

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