Theorem sbthlemi3 6797

Description: Lemma for isbth 6805. (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
sbthlemi3	|- ( (EXMID /\ ran g C_ A ) -> ( g " ( B \ ( f " U. D ) ) ) = ( A \ 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 sbthlemi3

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbthlem.1	. . . . . . 7 |- A e. _V
2		sbthlem.2	. . . . . . 7 |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
3	1, 2	sbthlem2 6796	. . . . . 6 |- ( ran g C_ A -> ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ U. D )
4	1, 2	sbthlem1 6795	. . . . . 6 |- U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) )
5	3, 4	jctil 308	. . . . 5 |- ( ran g C_ A -> ( U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) /\ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ U. D ) )
6		eqss 3076	. . . . 5 |- ( U. D = ( A \ ( g " ( B \ ( f " U. D ) ) ) ) <-> ( U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) /\ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ U. D ) )
7	5, 6	sylibr 133	. . . 4 |- ( ran g C_ A -> U. D = ( A \ ( g " ( B \ ( f " U. D ) ) ) ) )
8	7	difeq2d 3158	. . 3 |- ( ran g C_ A -> ( A \ U. D ) = ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) )
9	8	adantl 273	. 2 |- ( (EXMID /\ ran g C_ A ) -> ( A \ U. D ) = ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) )
10		imassrn 4848	. . . . 5 |- ( g " ( B \ ( f " U. D ) ) ) C_ ran g
11		sstr2 3068	. . . . 5 |- ( ( g " ( B \ ( f " U. D ) ) ) C_ ran g -> ( ran g C_ A -> ( g " ( B \ ( f " U. D ) ) ) C_ A ) )
12	10, 11	ax-mp 7	. . . 4 |- ( ran g C_ A -> ( g " ( B \ ( f " U. D ) ) ) C_ A )
13		exmidexmid 4078	. . . . . . 7 |- (EXMID -> DECID y e. ( g " ( B \ ( f " U. D ) ) ) )
14		dcstab 812	. . . . . . 7 |- (DECID y e. ( g " ( B \ ( f " U. D ) ) ) -> STAB y e. ( g " ( B \ ( f " U. D ) ) ) )
15	13, 14	syl 14	. . . . . 6 |- (EXMID -> STAB y e. ( g " ( B \ ( f " U. D ) ) ) )
16	15	alrimiv 1826	. . . . 5 |- (EXMID -> A. ySTAB y e. ( g " ( B \ ( f " U. D ) ) ) )
17		dfss4st 3273	. . . . 5 |- ( A. ySTAB y e. ( g " ( B \ ( f " U. D ) ) ) -> ( ( g " ( B \ ( f " U. D ) ) ) C_ A <-> ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) = ( g " ( B \ ( f " U. D ) ) ) ) )
18	16, 17	syl 14	. . . 4 |- (EXMID -> ( ( g " ( B \ ( f " U. D ) ) ) C_ A <-> ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) = ( g " ( B \ ( f " U. D ) ) ) ) )
19	12, 18	syl5ib 153	. . 3 |- (EXMID -> ( ran g C_ A -> ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) = ( g " ( B \ ( f " U. D ) ) ) ) )
20	19	imp 123	. 2 |- ( (EXMID /\ ran g C_ A ) -> ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) = ( g " ( B \ ( f " U. D ) ) ) )
21	9, 20	eqtr2d 2146	1 |- ( (EXMID /\ ran g C_ A ) -> ( g " ( B \ ( f " U. D ) ) ) = ( A \ U. D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104  STAB wstab 798  DECID wdc 802   A.wal 1310   = wceq 1312   e. wcel 1461   {cab 2099   _Vcvv 2655   \ cdif 3032   C_ wss 3035   U.cuni 3700  EXMIDwem 4076   ran crn 4498   "cima 4500

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-exmid 4077  df-xp 4503  df-cnv 4505  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510

This theorem is referenced by:  sbthlemi4  6798  sbthlemi5  6799