Theorem sbthlemi3 6948

Description: Lemma for isbth 6956. (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 6947	. . . . . 6 |- ( ran g C_ A -> ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ U. D )
4	1, 2	sbthlem1 6946	. . . . . 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 3168	. . . . 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 3251	. . 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 4974	. . . . 5 |- ( g " ( B \ ( f " U. D ) ) ) C_ ran g
11		sstr2 3160	. . . . 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 4191	. . . . . . 7 |- (EXMID -> DECID y e. ( g " ( B \ ( f " U. D ) ) ) )
14		dcstab 844	. . . . . . 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 1872	. . . . 5 |- (EXMID -> A. ySTAB y e. ( g " ( B \ ( f " U. D ) ) ) )
17		dfss4st 3366	. . . . 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 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 2209	1 |- ( (EXMID /\ ran g C_ A ) -> ( g " ( B \ ( f " U. D ) ) ) = ( A \ U. D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  STAB wstab 830  DECID wdc 834   A.wal 1351   = wceq 1353   e. wcel 2146   {cab 2161   _Vcvv 2735   \ cdif 3124   C_ wss 3127   U.cuni 3805  EXMIDwem 4189   ran crn 4621   "cima 4623

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-exmid 4190  df-xp 4626  df-cnv 4628  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633

This theorem is referenced by:  sbthlemi4  6949  sbthlemi5  6950