Theorem sbthlemi6 6939

Description: Lemma for isbth 6944. (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 524	. . 3 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> EXMID )
2		simprll 532	. . 3 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> dom g = B )
3		simprlr 533	. . 3 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran g C_ A )
4		simprr 527	. . 3 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' g )
5		rnun 5019	. . . . 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 4839	. . . . 5 |- ran H = ran ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
8		df-ima 4624	. . . . . 6 |- ( f " U. D ) = ran ( f |` U. D )
9	8	uneq1i 3277	. . . . 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 2201	. . . 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 6937	. . . . . 6 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
14		df-ima 4624	. . . . . 6 |- ( `' g " ( A \ U. D ) ) = ran ( `' g |` ( A \ U. D ) )
15	13, 14	eqtr3di 2218	. . . . 5 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( B \ ( f " U. D ) ) = ran ( `' g |` ( A \ U. D ) ) )
16	15	uneq2d 3281	. . . 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 2222	. . 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 1238	. 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 4964	. . . . . . 7 |- ( f " U. D ) C_ ran f
20		sstr2 3154	. . . . . . 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		undifdcss 6900	. . . . . . 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 4182	. . . . . . . . 9 |- (EXMID -> DECID y e. ( f " U. D ) )
25	24	ralrimivw 2544	. . . . . . . 8 |- (EXMID -> A. y e. B DECID y e. ( f " U. D ) )
26	25	biantrud 302	. . . . . . 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 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 2176	. . 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 2203	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 829   /\ w3a 973   = wceq 1348   e. wcel 2141   {cab 2156   A.wral 2448   _Vcvv 2730   \ cdif 3118   u. cun 3119   C_ wss 3121   U.cuni 3796  EXMIDwem 4180   `'ccnv 4610   dom cdm 4611   ran crn 4612   |` cres 4613   "cima 4614   Fun wfun 5192

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-exmid 4181  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-fun 5200

This theorem is referenced by:  sbthlemi9  6942