Theorem sbthlemi6 7064

Description: Lemma for isbth 7069. (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 527	. . 3 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> EXMID )
2		simprll 537	. . 3 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> dom g = B )
3		simprlr 538	. . 3 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran g C_ A )
4		simprr 531	. . 3 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' g )
5		rnun 5091	. . . . 5 |- ran ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) ) = ( ran ( f |` U. D ) u. ran ( `' g |` ( A \ U. D ) ) )
6		sbthlem.3	. . . . . 6 |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
7	6	rneqi 4906	. . . . 5 |- ran H = ran ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
8		df-ima 4688	. . . . . 6 |- ( f " U. D ) = ran ( f |` U. D )
9	8	uneq1i 3323	. . . . 5 |- ( ( f " U. D ) u. ran ( `' g |` ( A \ U. D ) ) ) = ( ran ( f |` U. D ) u. ran ( `' g |` ( A \ U. D ) ) )
10	5, 7, 9	3eqtr4i 2236	. . . 4 |- ran H = ( ( f " U. D ) u. ran ( `' g |` ( A \ U. D ) ) )
11		sbthlem.1	. . . . . . 7 |- A e. _V
12		sbthlem.2	. . . . . . 7 |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
13	11, 12	sbthlemi4 7062	. . . . . 6 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
14		df-ima 4688	. . . . . 6 |- ( `' g " ( A \ U. D ) ) = ran ( `' g |` ( A \ U. D ) )
15	13, 14	eqtr3di 2253	. . . . 5 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( B \ ( f " U. D ) ) = ran ( `' g |` ( A \ U. D ) ) )
16	15	uneq2d 3327	. . . 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 ) ) ) )
17	10, 16	eqtr4id 2257	. . 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 1255	. 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 5033	. . . . . . 7 |- ( f " U. D ) C_ ran f
20		sstr2 3200	. . . . . . 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 277	. . . . 5 |- ( (EXMID /\ ran f C_ B ) -> ( f " U. D ) C_ B )
23		undifdcss 7020	. . . . . . 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 ) ) )
24		exmidexmid 4240	. . . . . . . . 9 |- (EXMID -> DECID y e. ( f " U. D ) )
25	24	ralrimivw 2580	. . . . . . . 8 |- (EXMID -> A. y e. B DECID y e. ( f " U. D ) )
26	25	biantrud 304	. . . . . . 7 |- (EXMID -> ( ( f " U. D ) C_ B <-> ( ( f " U. D ) C_ B /\ A. y e. B DECID y e. ( f " U. D ) ) ) )
27	23, 26	bitr4id 199	. . . . . 6 |- (EXMID -> ( B = ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) <-> ( f " U. D ) C_ B ) )
28	27	adantr 276	. . . . 5 |- ( (EXMID /\ ran f C_ B ) -> ( B = ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) <-> ( f " U. D ) C_ B ) )
29	22, 28	mpbird 167	. . . 4 |- ( (EXMID /\ ran f C_ B ) -> B = ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) )
30	29	eqcomd 2211	. . 3 |- ( (EXMID /\ ran f C_ B ) -> ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) = B )
31	30	adantr 276	. 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 2238	1 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 836   /\ w3a 981   = wceq 1373   e. wcel 2176   {cab 2191   A.wral 2484   _Vcvv 2772   \ cdif 3163   u. cun 3164   C_ wss 3166   U.cuni 3850  EXMIDwem 4238   `'ccnv 4674   dom cdm 4675   ran crn 4676   |` cres 4677   "cima 4678   Fun wfun 5265

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-exmid 4239  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-fun 5273

This theorem is referenced by:  sbthlemi9  7067