Theorem sbthlem1 6934

Description: Lemma for isbth 6944. (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 3826	. 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 2281	. . . 4 |- ( x e. D <-> ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) )
4		difss2 3255	. . . . . . 7 |- ( ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) -> ( g " ( B \ ( f " x ) ) ) C_ A )
5		ssconb 3260	. . . . . . . 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 380	. . . . . . 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 123	. . . 4 |- ( ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) -> x C_ ( A \ ( g " ( B \ ( f " x ) ) ) ) )
10	3, 9	sylbi 120	. . 3 |- ( x e. D -> x C_ ( A \ ( g " ( B \ ( f " x ) ) ) ) )
11		elssuni 3824	. . . . 5 |- ( x e. D -> x C_ U. D )
12		imass2 4987	. . . . 5 |- ( x C_ U. D -> ( f " x ) C_ ( f " U. D ) )
13		sscon 3261	. . . . 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 4987	. . . 4 |- ( ( B \ ( f " U. D ) ) C_ ( B \ ( f " x ) ) -> ( g " ( B \ ( f " U. D ) ) ) C_ ( g " ( B \ ( f " x ) ) ) )
16		sscon 3261	. . . 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 3157	. 2 |- ( x e. D -> x C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) )
19	1, 18	mprgbir 2528	1 |- U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   {cab 2156   _Vcvv 2730   \ cdif 3118   C_ wss 3121   U.cuni 3796   "cima 4614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624

This theorem is referenced by:  sbthlem2  6935  sbthlemi3  6936  sbthlemi5  6938