Theorem sbthlemi4 7150

Description: Lemma for isbth 7157. (Contributed by NM, 27-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
sbthlemi4	|- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " 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 sbthlemi4

Step	Hyp	Ref	Expression
1		df-ima 4736	. 2 |- ( `' g " ( A \ U. D ) ) = ran ( `' g |` ( A \ U. D ) )
2		difss 3331	. . . . . . . 8 |- ( B \ ( f " U. D ) ) C_ B
3		sseq2 3249	. . . . . . . 8 |- ( dom g = B -> ( ( B \ ( f " U. D ) ) C_ dom g <-> ( B \ ( f " U. D ) ) C_ B ) )
4	2, 3	mpbiri 168	. . . . . . 7 |- ( dom g = B -> ( B \ ( f " U. D ) ) C_ dom g )
5		ssdmres 5033	. . . . . . 7 |- ( ( B \ ( f " U. D ) ) C_ dom g <-> dom ( g |` ( B \ ( f " U. D ) ) ) = ( B \ ( f " U. D ) ) )
6	4, 5	sylib 122	. . . . . 6 |- ( dom g = B -> dom ( g |` ( B \ ( f " U. D ) ) ) = ( B \ ( f " U. D ) ) )
7		dfdm4 4921	. . . . . 6 |- dom ( g |` ( B \ ( f " U. D ) ) ) = ran `' ( g |` ( B \ ( f " U. D ) ) )
8	6, 7	eqtr3di 2277	. . . . 5 |- ( dom g = B -> ( B \ ( f " U. D ) ) = ran `' ( g |` ( B \ ( f " U. D ) ) ) )
9	8	adantr 276	. . . 4 |- ( ( dom g = B /\ ran g C_ A ) -> ( B \ ( f " U. D ) ) = ran `' ( g |` ( B \ ( f " U. D ) ) ) )
10	9	3ad2ant2 1043	. . 3 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( B \ ( f " U. D ) ) = ran `' ( g |` ( B \ ( f " U. D ) ) ) )
11		funcnvres 5400	. . . . . . 7 |- ( Fun `' g -> `' ( g |` ( B \ ( f " U. D ) ) ) = ( `' g |` ( g " ( B \ ( f " U. D ) ) ) ) )
12	11	3ad2ant3 1044	. . . . . 6 |- ( (EXMID /\ ran g C_ A /\ Fun `' g ) -> `' ( g |` ( B \ ( f " U. D ) ) ) = ( `' g |` ( g " ( B \ ( f " U. D ) ) ) ) )
13		sbthlem.1	. . . . . . . . 9 |- A e. _V
14		sbthlem.2	. . . . . . . . 9 |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
15	13, 14	sbthlemi3 7149	. . . . . . . 8 |- ( (EXMID /\ ran g C_ A ) -> ( g " ( B \ ( f " U. D ) ) ) = ( A \ U. D ) )
16	15	reseq2d 5011	. . . . . . 7 |- ( (EXMID /\ ran g C_ A ) -> ( `' g |` ( g " ( B \ ( f " U. D ) ) ) ) = ( `' g |` ( A \ U. D ) ) )
17	16	3adant3 1041	. . . . . 6 |- ( (EXMID /\ ran g C_ A /\ Fun `' g ) -> ( `' g |` ( g " ( B \ ( f " U. D ) ) ) ) = ( `' g |` ( A \ U. D ) ) )
18	12, 17	eqtrd 2262	. . . . 5 |- ( (EXMID /\ ran g C_ A /\ Fun `' g ) -> `' ( g |` ( B \ ( f " U. D ) ) ) = ( `' g |` ( A \ U. D ) ) )
19	18	rneqd 4959	. . . 4 |- ( (EXMID /\ ran g C_ A /\ Fun `' g ) -> ran `' ( g |` ( B \ ( f " U. D ) ) ) = ran ( `' g |` ( A \ U. D ) ) )
20	19	3adant2l 1256	. . 3 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ran `' ( g |` ( B \ ( f " U. D ) ) ) = ran ( `' g |` ( A \ U. D ) ) )
21	10, 20	eqtrd 2262	. 2 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( B \ ( f " U. D ) ) = ran ( `' g |` ( A \ U. D ) ) )
22	1, 21	eqtr4id 2281	1 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   {cab 2215   _Vcvv 2800   \ cdif 3195   C_ wss 3198   U.cuni 3891  EXMIDwem 4282   `'ccnv 4722   dom cdm 4723   ran crn 4724   |` cres 4725   "cima 4726   Fun wfun 5318

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-exmid 4283  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-fun 5326

This theorem is referenced by:  sbthlemi6  7152  sbthlemi8  7154