Theorem sbthlemi4 6848

Description: Lemma for isbth 6855. (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 4731	. . . . . 6 |- dom ( g |` ( B \ ( f " U. D ) ) ) = ran `' ( g |` ( B \ ( f " U. D ) ) )
2		difss 3202	. . . . . . . 8 |- ( B \ ( f " U. D ) ) C_ B
3		sseq2 3121	. . . . . . . 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 4841	. . . . . . 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 2187	. . . . 5 |- ( dom g = B -> ( B \ ( f " U. D ) ) = ran `' ( g |` ( B \ ( f " U. D ) ) ) )
8	7	adantr 274	. . . 4 |- ( ( dom g = B /\ ran g C_ A ) -> ( B \ ( f " U. D ) ) = ran `' ( g |` ( B \ ( f " U. D ) ) ) )
9	8	3ad2ant2 1003	. . 3 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( B \ ( f " U. D ) ) = ran `' ( g |` ( B \ ( f " U. D ) ) ) )
10		funcnvres 5196	. . . . . . 7 |- ( Fun `' g -> `' ( g |` ( B \ ( f " U. D ) ) ) = ( `' g |` ( g " ( B \ ( f " U. D ) ) ) ) )
11	10	3ad2ant3 1004	. . . . . 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 6847	. . . . . . . 8 |- ( (EXMID /\ ran g C_ A ) -> ( g " ( B \ ( f " U. D ) ) ) = ( A \ U. D ) )
15	14	reseq2d 4819	. . . . . . 7 |- ( (EXMID /\ ran g C_ A ) -> ( `' g |` ( g " ( B \ ( f " U. D ) ) ) ) = ( `' g |` ( A \ U. D ) ) )
16	15	3adant3 1001	. . . . . 6 |- ( (EXMID /\ ran g C_ A /\ Fun `' g ) -> ( `' g |` ( g " ( B \ ( f " U. D ) ) ) ) = ( `' g |` ( A \ U. D ) ) )
17	11, 16	eqtrd 2172	. . . . 5 |- ( (EXMID /\ ran g C_ A /\ Fun `' g ) -> `' ( g |` ( B \ ( f " U. D ) ) ) = ( `' g |` ( A \ U. D ) ) )
18	17	rneqd 4768	. . . 4 |- ( (EXMID /\ ran g C_ A /\ Fun `' g ) -> ran `' ( g |` ( B \ ( f " U. D ) ) ) = ran ( `' g |` ( A \ U. D ) ) )
19	18	3adant2l 1210	. . 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 2172	. 2 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( B \ ( f " U. D ) ) = ran ( `' g |` ( A \ U. D ) ) )
21		df-ima 4552	. 2 |- ( `' g " ( A \ U. D ) ) = ran ( `' g |` ( A \ U. D ) )
22	20, 21	syl6reqr 2191	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 962   = wceq 1331   e. wcel 1480   {cab 2125   _Vcvv 2686   \ cdif 3068   C_ wss 3071   U.cuni 3736  EXMIDwem 4118   `'ccnv 4538   dom cdm 4539   ran crn 4540   |` cres 4541   "cima 4542   Fun wfun 5117

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-exmid 4119  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-fun 5125

This theorem is referenced by:  sbthlemi6  6850  sbthlemi8  6852