Theorem sbthlemi6 7021

Description: Lemma for isbth 7026. (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 5074	. . . . 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 4890	. . . . 5 |- ran H = ran ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
8		df-ima 4672	. . . . . 6 |- ( f " U. D ) = ran ( f |` U. D )
9	8	uneq1i 3309	. . . . 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 2224	. . . 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 7019	. . . . . 6 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
14		df-ima 4672	. . . . . 6 |- ( `' g " ( A \ U. D ) ) = ran ( `' g |` ( A \ U. D ) )
15	13, 14	eqtr3di 2241	. . . . 5 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( B \ ( f " U. D ) ) = ran ( `' g |` ( A \ U. D ) ) )
16	15	uneq2d 3313	. . . 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 2245	. . 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 1254	. 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 5016	. . . . . . 7 |- ( f " U. D ) C_ ran f
20		sstr2 3186	. . . . . . 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 6979	. . . . . . 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 4225	. . . . . . . . 9 |- (EXMID -> DECID y e. ( f " U. D ) )
25	24	ralrimivw 2568	. . . . . . . 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 2199	. . 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 2226	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 835   /\ w3a 980   = wceq 1364   e. wcel 2164   {cab 2179   A.wral 2472   _Vcvv 2760   \ cdif 3150   u. cun 3151   C_ wss 3153   U.cuni 3835  EXMIDwem 4223   `'ccnv 4658   dom cdm 4659   ran crn 4660   |` cres 4661   "cima 4662   Fun wfun 5248

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-exmid 4224  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-fun 5256

This theorem is referenced by:  sbthlemi9  7024