Theorem sbthlemi3 6915

Description: Lemma for isbth 6923. (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 6914	. . . . . 6 |- ( ran g C_ A -> ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ U. D )
4	1, 2	sbthlem1 6913	. . . . . 6 |- U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) )
5	3, 4	jctil 310	. . . . 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 3152	. . . . 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 3235	. . 3 |- ( ran g C_ A -> ( A \ U. D ) = ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) )
9	8	adantl 275	. 2 |- ( (EXMID /\ ran g C_ A ) -> ( A \ U. D ) = ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) )
10		imassrn 4951	. . . . 5 |- ( g " ( B \ ( f " U. D ) ) ) C_ ran g
11		sstr2 3144	. . . . 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 4169	. . . . . . 7 |- (EXMID -> DECID y e. ( g " ( B \ ( f " U. D ) ) ) )
14		dcstab 834	. . . . . . 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 1861	. . . . 5 |- (EXMID -> A. ySTAB y e. ( g " ( B \ ( f " U. D ) ) ) )
17		dfss4st 3350	. . . . 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 2198	1 |- ( (EXMID /\ ran g C_ A ) -> ( g " ( B \ ( f " U. D ) ) ) = ( A \ U. D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104  STAB wstab 820  DECID wdc 824   A.wal 1340   = wceq 1342   e. wcel 2135   {cab 2150   _Vcvv 2721   \ cdif 3108   C_ wss 3111   U.cuni 3783  EXMIDwem 4167   ran crn 4599   "cima 4601

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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-exmid 4168  df-xp 4604  df-cnv 4606  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611

This theorem is referenced by:  sbthlemi4  6916  sbthlemi5  6917