Theorem sbthlemi6 6850

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 ) ) }
sbthlem.3	|- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )

Assertion

Ref	Expression
sbthlemi6	|- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B )

Distinct variable groups:   x, A   x, B   x, D   x, f   x, g   x, H

Allowed substitution hints:   A( f, g)   B( f, g)   D( f, g)   H( f, g)

Proof of Theorem sbthlemi6

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 518	. . 3 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> EXMID )
2		simprll 526	. . 3 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> dom g = B )
3		simprlr 527	. . 3 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran g C_ A )
4		simprr 521	. . 3 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' g )
5		df-ima 4552	. . . . . 6 |- ( `' g " ( A \ U. D ) ) = ran ( `' g |` ( A \ U. D ) )
6		sbthlem.1	. . . . . . 7 |- A e. _V
7		sbthlem.2	. . . . . . 7 |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
8	6, 7	sbthlemi4 6848	. . . . . 6 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
9	5, 8	syl5reqr 2187	. . . . 5 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( B \ ( f " U. D ) ) = ran ( `' g |` ( A \ U. D ) ) )
10	9	uneq2d 3230	. . . 4 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) = ( ( f " U. D ) u. ran ( `' g |` ( A \ U. D ) ) ) )
11		rnun 4947	. . . . 5 |- ran ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) ) = ( ran ( f |` U. D ) u. ran ( `' g |` ( A \ U. D ) ) )
12		sbthlem.3	. . . . . 6 |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
13	12	rneqi 4767	. . . . 5 |- ran H = ran ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
14		df-ima 4552	. . . . . 6 |- ( f " U. D ) = ran ( f |` U. D )
15	14	uneq1i 3226	. . . . 5 |- ( ( f " U. D ) u. ran ( `' g |` ( A \ U. D ) ) ) = ( ran ( f |` U. D ) u. ran ( `' g |` ( A \ U. D ) ) )
16	11, 13, 15	3eqtr4i 2170	. . . 4 |- ran H = ( ( f " U. D ) u. ran ( `' g |` ( A \ U. D ) ) )
17	10, 16	syl6reqr 2191	. . 3 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ran H = ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) )
18	1, 2, 3, 4, 17	syl121anc 1221	. 2 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) )
19		imassrn 4892	. . . . . . 7 |- ( f " U. D ) C_ ran f
20		sstr2 3104	. . . . . . 7 |- ( ( f " U. D ) C_ ran f -> ( ran f C_ B -> ( f " U. D ) C_ B ) )
21	19, 20	ax-mp 5	. . . . . 6 |- ( ran f C_ B -> ( f " U. D ) C_ B )
22	21	adantl 275	. . . . 5 |- ( (EXMID /\ ran f C_ B ) -> ( f " U. D ) C_ B )
23		exmidexmid 4120	. . . . . . . . 9 |- (EXMID -> DECID y e. ( f " U. D ) )
24	23	ralrimivw 2506	. . . . . . . 8 |- (EXMID -> A. y e. B DECID y e. ( f " U. D ) )
25	24	biantrud 302	. . . . . . 7 |- (EXMID -> ( ( f " U. D ) C_ B <-> ( ( f " U. D ) C_ B /\ A. y e. B DECID y e. ( f " U. D ) ) ) )
26		undifdcss 6811	. . . . . . 7 |- ( B = ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) <-> ( ( f " U. D ) C_ B /\ A. y e. B DECID y e. ( f " U. D ) ) )
27	25, 26	syl6rbbr 198	. . . . . 6 |- (EXMID -> ( B = ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) <-> ( f " U. D ) C_ B ) )
28	27	adantr 274	. . . . 5 |- ( (EXMID /\ ran f C_ B ) -> ( B = ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) <-> ( f " U. D ) C_ B ) )
29	22, 28	mpbird 166	. . . 4 |- ( (EXMID /\ ran f C_ B ) -> B = ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) )
30	29	eqcomd 2145	. . 3 |- ( (EXMID /\ ran f C_ B ) -> ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) = B )
31	30	adantr 274	. 2 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) = B )
32	18, 31	eqtrd 2172	1 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104  DECID wdc 819   /\ w3a 962   = wceq 1331   e. wcel 1480   {cab 2125   A.wral 2416   _Vcvv 2686   \ cdif 3068   u. cun 3069   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:  sbthlemi9  6853