Theorem sbthlemi4 6776

Description: Lemma for isbth 6783. (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		dfdm4 4669	. . . . . 6 |- dom ( g |` ( B \ ( f " U. D ) ) ) = ran `' ( g |` ( B \ ( f " U. D ) ) )
2		difss 3149	. . . . . . . 8 |- ( B \ ( f " U. D ) ) C_ B
3		sseq2 3071	. . . . . . . 8 |- ( dom g = B -> ( ( B \ ( f " U. D ) ) C_ dom g <-> ( B \ ( f " U. D ) ) C_ B ) )
4	2, 3	mpbiri 167	. . . . . . 7 |- ( dom g = B -> ( B \ ( f " U. D ) ) C_ dom g )
5		ssdmres 4777	. . . . . . 7 |- ( ( B \ ( f " U. D ) ) C_ dom g <-> dom ( g |` ( B \ ( f " U. D ) ) ) = ( B \ ( f " U. D ) ) )
6	4, 5	sylib 121	. . . . . 6 |- ( dom g = B -> dom ( g |` ( B \ ( f " U. D ) ) ) = ( B \ ( f " U. D ) ) )
7	1, 6	syl5reqr 2147	. . . . 5 |- ( dom g = B -> ( B \ ( f " U. D ) ) = ran `' ( g |` ( B \ ( f " U. D ) ) ) )
8	7	adantr 272	. . . 4 |- ( ( dom g = B /\ ran g C_ A ) -> ( B \ ( f " U. D ) ) = ran `' ( g |` ( B \ ( f " U. D ) ) ) )
9	8	3ad2ant2 971	. . 3 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( B \ ( f " U. D ) ) = ran `' ( g |` ( B \ ( f " U. D ) ) ) )
10		funcnvres 5132	. . . . . . 7 |- ( Fun `' g -> `' ( g |` ( B \ ( f " U. D ) ) ) = ( `' g |` ( g " ( B \ ( f " U. D ) ) ) ) )
11	10	3ad2ant3 972	. . . . . 6 |- ( (EXMID /\ ran g C_ A /\ Fun `' g ) -> `' ( g |` ( B \ ( f " U. D ) ) ) = ( `' g |` ( g " ( B \ ( f " U. D ) ) ) ) )
12		sbthlem.1	. . . . . . . . 9 |- A e. _V
13		sbthlem.2	. . . . . . . . 9 |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
14	12, 13	sbthlemi3 6775	. . . . . . . 8 |- ( (EXMID /\ ran g C_ A ) -> ( g " ( B \ ( f " U. D ) ) ) = ( A \ U. D ) )
15	14	reseq2d 4755	. . . . . . 7 |- ( (EXMID /\ ran g C_ A ) -> ( `' g |` ( g " ( B \ ( f " U. D ) ) ) ) = ( `' g |` ( A \ U. D ) ) )
16	15	3adant3 969	. . . . . 6 |- ( (EXMID /\ ran g C_ A /\ Fun `' g ) -> ( `' g |` ( g " ( B \ ( f " U. D ) ) ) ) = ( `' g |` ( A \ U. D ) ) )
17	11, 16	eqtrd 2132	. . . . 5 |- ( (EXMID /\ ran g C_ A /\ Fun `' g ) -> `' ( g |` ( B \ ( f " U. D ) ) ) = ( `' g |` ( A \ U. D ) ) )
18	17	rneqd 4706	. . . 4 |- ( (EXMID /\ ran g C_ A /\ Fun `' g ) -> ran `' ( g |` ( B \ ( f " U. D ) ) ) = ran ( `' g |` ( A \ U. D ) ) )
19	18	3adant2l 1178	. . 3 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ran `' ( g |` ( B \ ( f " U. D ) ) ) = ran ( `' g |` ( A \ U. D ) ) )
20	9, 19	eqtrd 2132	. 2 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( B \ ( f " U. D ) ) = ran ( `' g |` ( A \ U. D ) ) )
21		df-ima 4490	. 2 |- ( `' g " ( A \ U. D ) ) = ran ( `' g |` ( A \ U. D ) )
22	20, 21	syl6reqr 2151	1 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 930   = wceq 1299   e. wcel 1448   {cab 2086   _Vcvv 2641   \ cdif 3018   C_ wss 3021   U.cuni 3683  EXMIDwem 4058   `'ccnv 4476   dom cdm 4477   ran crn 4478   |` cres 4479   "cima 4480   Fun wfun 5053

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-stab 782  df-dc 787  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-exmid 4059  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-fun 5061

This theorem is referenced by:  sbthlemi6  6778  sbthlemi8  6780