Theorem sbthlem1 7240

Description: Lemma for isbth 7250. (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 3949	. 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 2345	. . . 4 |- ( x e. D <-> ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) )
4		difss2 3351	. . . . . . 7 |- ( ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) -> ( g " ( B \ ( f " x ) ) ) C_ A )
5		ssconb 3356	. . . . . . . 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 3947	. . . . 5 |- ( x e. D -> x C_ U. D )
12		imass2 5143	. . . . 5 |- ( x C_ U. D -> ( f " x ) C_ ( f " U. D ) )
13		sscon 3357	. . . . 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 5143	. . . 4 |- ( ( B \ ( f " U. D ) ) C_ ( B \ ( f " x ) ) -> ( g " ( B \ ( f " U. D ) ) ) C_ ( g " ( B \ ( f " x ) ) ) )
16		sscon 3357	. . . 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 3252	. 2 |- ( x e. D -> x C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) )
19	1, 18	mprgbir 2602	1 |- U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   {cab 2220   _Vcvv 2815   \ cdif 3211   C_ wss 3214   U.cuni 3919   "cima 4757

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-xp 4760  df-cnv 4762  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767

This theorem is referenced by:  sbthlem2  7241  sbthlemi3  7242  sbthlemi5  7244