Theorem sbthlemi3 7025

Description: Lemma for isbth 7033. (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 7024	. . . . . 6 |- ( ran g C_ A -> ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ U. D )
4	1, 2	sbthlem1 7023	. . . . . 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 3198	. . . . 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 3281	. . 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 5020	. . . . 5 |- ( g " ( B \ ( f " U. D ) ) ) C_ ran g
11		sstr2 3190	. . . . 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 4229	. . . . . . 7 |- (EXMID -> DECID y e. ( g " ( B \ ( f " U. D ) ) ) )
14		dcstab 845	. . . . . . 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 1888	. . . . 5 |- (EXMID -> A. ySTAB y e. ( g " ( B \ ( f " U. D ) ) ) )
17		dfss4st 3396	. . . . 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 2230	1 |- ( (EXMID /\ ran g C_ A ) -> ( g " ( B \ ( f " U. D ) ) ) = ( A \ U. D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  STAB wstab 831  DECID wdc 835   A.wal 1362   = wceq 1364   e. wcel 2167   {cab 2182   _Vcvv 2763   \ cdif 3154   C_ wss 3157   U.cuni 3839  EXMIDwem 4227   ran crn 4664   "cima 4666

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-exmid 4228  df-xp 4669  df-cnv 4671  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676

This theorem is referenced by:  sbthlemi4  7026  sbthlemi5  7027