Theorem sbthlemi3 7087

Description: Lemma for isbth 7095. (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 7086	. . . . . 6 |- ( ran g C_ A -> ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ U. D )
4	1, 2	sbthlem1 7085	. . . . . 6 |- U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) )
5	3, 4	jctil 312	. . . . 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 3216	. . . . 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 134	. . . 4 |- ( ran g C_ A -> U. D = ( A \ ( g " ( B \ ( f " U. D ) ) ) ) )
8	7	difeq2d 3299	. . 3 |- ( ran g C_ A -> ( A \ U. D ) = ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) )
9	8	adantl 277	. 2 |- ( (EXMID /\ ran g C_ A ) -> ( A \ U. D ) = ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) )
10		imassrn 5052	. . . . 5 |- ( g " ( B \ ( f " U. D ) ) ) C_ ran g
11		sstr2 3208	. . . . 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 5	. . . 4 |- ( ran g C_ A -> ( g " ( B \ ( f " U. D ) ) ) C_ A )
13		exmidexmid 4256	. . . . . . 7 |- (EXMID -> DECID y e. ( g " ( B \ ( f " U. D ) ) ) )
14		dcstab 846	. . . . . . 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 1898	. . . . 5 |- (EXMID -> A. ySTAB y e. ( g " ( B \ ( f " U. D ) ) ) )
17		dfss4st 3414	. . . . 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	imbitrid 154	. . 3 |- (EXMID -> ( ran g C_ A -> ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) = ( g " ( B \ ( f " U. D ) ) ) ) )
20	19	imp 124	. 2 |- ( (EXMID /\ ran g C_ A ) -> ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) = ( g " ( B \ ( f " U. D ) ) ) )
21	9, 20	eqtr2d 2241	1 |- ( (EXMID /\ ran g C_ A ) -> ( g " ( B \ ( f " U. D ) ) ) = ( A \ U. D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  STAB wstab 832  DECID wdc 836   A.wal 1371   = wceq 1373   e. wcel 2178   {cab 2193   _Vcvv 2776   \ cdif 3171   C_ wss 3174   U.cuni 3864  EXMIDwem 4254   ran crn 4694   "cima 4696

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-exmid 4255  df-xp 4699  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706

This theorem is referenced by:  sbthlemi4  7088  sbthlemi5  7089