Theorem sbthlemi4 7158

Description: Lemma for isbth 7165. (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 4738	. 2 |- ( `' g " ( A \ U. D ) ) = ran ( `' g |` ( A \ U. D ) )
2		difss 3333	. . . . . . . 8 |- ( B \ ( f " U. D ) ) C_ B
3		sseq2 3251	. . . . . . . 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 5035	. . . . . . 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 4923	. . . . . 6 |- dom ( g |` ( B \ ( f " U. D ) ) ) = ran `' ( g |` ( B \ ( f " U. D ) ) )
8	6, 7	eqtr3di 2279	. . . . 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 1045	. . 3 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( B \ ( f " U. D ) ) = ran `' ( g |` ( B \ ( f " U. D ) ) ) )
11		funcnvres 5403	. . . . . . 7 |- ( Fun `' g -> `' ( g |` ( B \ ( f " U. D ) ) ) = ( `' g |` ( g " ( B \ ( f " U. D ) ) ) ) )
12	11	3ad2ant3 1046	. . . . . 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 7157	. . . . . . . 8 |- ( (EXMID /\ ran g C_ A ) -> ( g " ( B \ ( f " U. D ) ) ) = ( A \ U. D ) )
16	15	reseq2d 5013	. . . . . . 7 |- ( (EXMID /\ ran g C_ A ) -> ( `' g |` ( g " ( B \ ( f " U. D ) ) ) ) = ( `' g |` ( A \ U. D ) ) )
17	16	3adant3 1043	. . . . . 6 |- ( (EXMID /\ ran g C_ A /\ Fun `' g ) -> ( `' g |` ( g " ( B \ ( f " U. D ) ) ) ) = ( `' g |` ( A \ U. D ) ) )
18	12, 17	eqtrd 2264	. . . . 5 |- ( (EXMID /\ ran g C_ A /\ Fun `' g ) -> `' ( g |` ( B \ ( f " U. D ) ) ) = ( `' g |` ( A \ U. D ) ) )
19	18	rneqd 4961	. . . 4 |- ( (EXMID /\ ran g C_ A /\ Fun `' g ) -> ran `' ( g |` ( B \ ( f " U. D ) ) ) = ran ( `' g |` ( A \ U. D ) ) )
20	19	3adant2l 1258	. . 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 2264	. 2 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( B \ ( f " U. D ) ) = ran ( `' g |` ( A \ U. D ) ) )
22	1, 21	eqtr4id 2283	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 1004   = wceq 1397   e. wcel 2202   {cab 2217   _Vcvv 2802   \ cdif 3197   C_ wss 3200   U.cuni 3893  EXMIDwem 4284   `'ccnv 4724   dom cdm 4725   ran crn 4726   |` cres 4727   "cima 4728   Fun wfun 5320

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-exmid 4285  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-fun 5328

This theorem is referenced by:  sbthlemi6  7160  sbthlemi8  7162