Theorem sbthlem1 5510

Description: Lemma for sbth 5520.

Hypotheses

Ref	Expression
sbthlem.1	|- A e. _V
sbthlem.2	|- D = {x | (x C_ A /\ (g"(B \ (f"x))) C_ (A \ x))}

Assertion

Ref	Expression
sbthlem1	|- U.D C_ (A \ (g"(B \ (f"U.D))))

Distinct variable groups:   x,A   x,B   x,D   x,f   x,g

Proof of Theorem sbthlem1

Step	Hyp	Ref	Expression
1		unissb 3208	. 2 |- (U.D C_ (A \ (g"(B \ (f"U.D)))) <-> A.x e. D x C_ (A \ (g"(B \ (f"U.D)))))
2		sbthlem.2	. . . . 5 |- D = {x | (x C_ A /\ (g"(B \ (f"x))) C_ (A \ x))}
3	2	abeq2i 2001	. . . 4 |- (x e. D <-> (x C_ A /\ (g"(B \ (f"x))) C_ (A \ x)))
4		ssconb 2738	. . . . . . . 8 |- ((x C_ A /\ (g"(B \ (f"x))) C_ A) -> (x C_ (A \ (g"(B \ (f"x)))) <-> (g"(B \ (f"x))) C_ (A \ x)))
5	4	exbiri 421	. . . . . . 7 |- (x C_ A -> ((g"(B \ (f"x))) C_ A -> ((g"(B \ (f"x))) C_ (A \ x) -> x C_ (A \ (g"(B \ (f"x)))))))
6		difss 2735	. . . . . . . 8 |- (A \ x) C_ A
7		sstr2 2623	. . . . . . . 8 |- ((g"(B \ (f"x))) C_ (A \ x) -> ((A \ x) C_ A -> (g"(B \ (f"x))) C_ A))
8	6, 7	mpi 55	. . . . . . 7 |- ((g"(B \ (f"x))) C_ (A \ x) -> (g"(B \ (f"x))) C_ A)
9	5, 8	syl5 20	. . . . . 6 |- (x C_ A -> ((g"(B \ (f"x))) C_ (A \ x) -> ((g"(B \ (f"x))) C_ (A \ x) -> x C_ (A \ (g"(B \ (f"x)))))))
10	9	pm2.43d 79	. . . . 5 |- (x C_ A -> ((g"(B \ (f"x))) C_ (A \ x) -> x C_ (A \ (g"(B \ (f"x))))))
11	10	imp 377	. . . 4 |- ((x C_ A /\ (g"(B \ (f"x))) C_ (A \ x)) -> x C_ (A \ (g"(B \ (f"x)))))
12	3, 11	sylbi 216	. . 3 |- (x e. D -> x C_ (A \ (g"(B \ (f"x)))))
13		elssuni 3206	. . . . 5 |- (x e. D -> x C_ U.D)
14		imass2 4299	. . . . 5 |- (x C_ U.D -> (f"x) C_ (f"U.D))
15		sscon 2739	. . . . 5 |- ((f"x) C_ (f"U.D) -> (B \ (f"U.D)) C_ (B \ (f"x)))
16	13, 14, 15	3syl 24	. . . 4 |- (x e. D -> (B \ (f"U.D)) C_ (B \ (f"x)))
17		imass2 4299	. . . 4 |- ((B \ (f"U.D)) C_ (B \ (f"x)) -> (g"(B \ (f"U.D))) C_ (g"(B \ (f"x))))
18		sscon 2739	. . . 4 |- ((g"(B \ (f"U.D))) C_ (g"(B \ (f"x))) -> (A \ (g"(B \ (f"x)))) C_ (A \ (g"(B \ (f"U.D)))))
19	16, 17, 18	3syl 24	. . 3 |- (x e. D -> (A \ (g"(B \ (f"x)))) C_ (A \ (g"(B \ (f"U.D)))))
20	12, 19	sstrd 2627	. 2 |- (x e. D -> x C_ (A \ (g"(B \ (f"U.D)))))
21	1, 20	mprgbir 2163	1 |- U.D C_ (A \ (g"(B \ (f"U.D))))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292   \ cdif 2590   C_ wss 2593  U.cuni 3177  "cima 3989

This theorem is referenced by:  sbthlem2 5511  sbthlem3 5512  sbthlem5 5514

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-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-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007

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