Theorem sbthlem1 8070

Description: Lemma for sbth 8080. (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 4469	. 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 2735	. . . 4 |- ( x e. D <-> ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) )
4		difss2 3739	. . . . . . 7 |- ( ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) -> ( g " ( B \ ( f " x ) ) ) C_ A )
5		ssconb 3743	. . . . . . . 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 652	. . . . . . 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 34	. . . . . 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 53	. . . . 5 |- ( x C_ A -> ( ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) -> x C_ ( A \ ( g " ( B \ ( f " x ) ) ) ) ) )
9	8	imp 445	. . . 4 |- ( ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) -> x C_ ( A \ ( g " ( B \ ( f " x ) ) ) ) )
10	3, 9	sylbi 207	. . 3 |- ( x e. D -> x C_ ( A \ ( g " ( B \ ( f " x ) ) ) ) )
11		elssuni 4467	. . . . 5 |- ( x e. D -> x C_ U. D )
12		imass2 5501	. . . . 5 |- ( x C_ U. D -> ( f " x ) C_ ( f " U. D ) )
13		sscon 3744	. . . . 5 |- ( ( f " x ) C_ ( f " U. D ) -> ( B \ ( f " U. D ) ) C_ ( B \ ( f " x ) ) )
14	11, 12, 13	3syl 18	. . . 4 |- ( x e. D -> ( B \ ( f " U. D ) ) C_ ( B \ ( f " x ) ) )
15		imass2 5501	. . . 4 |- ( ( B \ ( f " U. D ) ) C_ ( B \ ( f " x ) ) -> ( g " ( B \ ( f " U. D ) ) ) C_ ( g " ( B \ ( f " x ) ) ) )
16		sscon 3744	. . . 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 18	. . 3 |- ( x e. D -> ( A \ ( g " ( B \ ( f " x ) ) ) ) C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) )
18	10, 17	sstrd 3613	. 2 |- ( x e. D -> x C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) )
19	1, 18	mprgbir 2927	1 |- U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   \ cdif 3571   C_ wss 3574   U.cuni 4436   "cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  sbthlem2  8071  sbthlem3  8072  sbthlem5  8074