Theorem sbthlem1 7016

Description: Lemma for isbth 7026. (Contributed by NM, 22-Mar-1998.)

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

Allowed substitution hints:   A( f, g)   B( f, g)   D( f, g)

Proof of Theorem sbthlem1

Step	Hyp	Ref	Expression
1		unissb 3865	. 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 2304	. . . 4 |- ( x e. D <-> ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) )
4		difss2 3287	. . . . . . 7 |- ( ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) -> ( g " ( B \ ( f " x ) ) ) C_ A )
5		ssconb 3292	. . . . . . . 8 |- ( ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ A ) -> ( x C_ ( A \ ( g " ( B \ ( f " x ) ) ) ) <-> ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) )
6	5	exbiri 382	. . . . . . 7 |- ( x C_ A -> ( ( g " ( B \ ( f " x ) ) ) C_ A -> ( ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) -> x C_ ( A \ ( g " ( B \ ( f " x ) ) ) ) ) ) )
7	4, 6	syl5 32	. . . . . 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 ) ) ) ) ) ) )
8	7	pm2.43d 50	. . . . 5 |- ( x C_ A -> ( ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) -> x C_ ( A \ ( g " ( B \ ( f " x ) ) ) ) ) )
9	8	imp 124	. . . 4 |- ( ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) -> x C_ ( A \ ( g " ( B \ ( f " x ) ) ) ) )
10	3, 9	sylbi 121	. . 3 |- ( x e. D -> x C_ ( A \ ( g " ( B \ ( f " x ) ) ) ) )
11		elssuni 3863	. . . . 5 |- ( x e. D -> x C_ U. D )
12		imass2 5041	. . . . 5 |- ( x C_ U. D -> ( f " x ) C_ ( f " U. D ) )
13		sscon 3293	. . . . 5 |- ( ( f " x ) C_ ( f " U. D ) -> ( B \ ( f " U. D ) ) C_ ( B \ ( f " x ) ) )
14	11, 12, 13	3syl 17	. . . 4 |- ( x e. D -> ( B \ ( f " U. D ) ) C_ ( B \ ( f " x ) ) )
15		imass2 5041	. . . 4 |- ( ( B \ ( f " U. D ) ) C_ ( B \ ( f " x ) ) -> ( g " ( B \ ( f " U. D ) ) ) C_ ( g " ( B \ ( f " x ) ) ) )
16		sscon 3293	. . . 4 |- ( ( g " ( B \ ( f " U. D ) ) ) C_ ( g " ( B \ ( f " x ) ) ) -> ( A \ ( g " ( B \ ( f " x ) ) ) ) C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) )
17	14, 15, 16	3syl 17	. . 3 |- ( x e. D -> ( A \ ( g " ( B \ ( f " x ) ) ) ) C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) )
18	10, 17	sstrd 3189	. 2 |- ( x e. D -> x C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) )
19	1, 18	mprgbir 2552	1 |- U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   {cab 2179   _Vcvv 2760   \ cdif 3150   C_ wss 3153   U.cuni 3835   "cima 4662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-cnv 4667  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672

This theorem is referenced by:  sbthlem2  7017  sbthlemi3  7018  sbthlemi5  7020