Theorem sbthlemi8 6852

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
sbthlemi8	|- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' H )

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 sbthlemi8

Step	Hyp	Ref	Expression
1		funres11 5195	. . . 4 |- ( Fun `' f -> Fun `' ( f |` U. D ) )
2	1	ad2antlr 480	. . 3 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' ( f |` U. D ) )
3		funcnvcnv 5182	. . . . . 6 |- ( Fun g -> Fun `' `' g )
4		funres11 5195	. . . . . 6 |- ( Fun `' `' g -> Fun `' ( `' g |` ( A \ U. D ) ) )
5	3, 4	syl 14	. . . . 5 |- ( Fun g -> Fun `' ( `' g |` ( A \ U. D ) ) )
6	5	ad2antrr 479	. . . 4 |- ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) -> Fun `' ( `' g |` ( A \ U. D ) ) )
7	6	ad2antrl 481	. . 3 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' ( `' g |` ( A \ U. D ) ) )
8		simpll 518	. . . 4 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> EXMID )
9		simprll 526	. . . . 5 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ( Fun g /\ dom g = B ) )
10	9	simprd 113	. . . 4 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> dom g = B )
11		simprlr 527	. . . 4 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ran g C_ A )
12		simprr 521	. . . 4 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' g )
13		df-ima 4552	. . . . . . 7 |- ( f " U. D ) = ran ( f |` U. D )
14		df-rn 4550	. . . . . . 7 |- ran ( f |` U. D ) = dom `' ( f |` U. D )
15	13, 14	eqtr2i 2161	. . . . . 6 |- dom `' ( f |` U. D ) = ( f " U. D )
16		df-ima 4552	. . . . . . . 8 |- ( `' g " ( A \ U. D ) ) = ran ( `' g |` ( A \ U. D ) )
17		df-rn 4550	. . . . . . . 8 |- ran ( `' g |` ( A \ U. D ) ) = dom `' ( `' g |` ( A \ U. D ) )
18	16, 17	eqtri 2160	. . . . . . 7 |- ( `' g " ( A \ U. D ) ) = dom `' ( `' g |` ( A \ U. D ) )
19		sbthlem.1	. . . . . . . 8 |- A e. _V
20		sbthlem.2	. . . . . . . 8 |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
21	19, 20	sbthlemi4 6848	. . . . . . 7 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
22	18, 21	syl5eqr 2186	. . . . . 6 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> dom `' ( `' g |` ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
23		ineq12 3272	. . . . . 6 |- ( ( dom `' ( f |` U. D ) = ( f " U. D ) /\ dom `' ( `' g |` ( A \ U. D ) ) = ( B \ ( f " U. D ) ) ) -> ( dom `' ( f |` U. D ) i^i dom `' ( `' g |` ( A \ U. D ) ) ) = ( ( f " U. D ) i^i ( B \ ( f " U. D ) ) ) )
24	15, 22, 23	sylancr 410	. . . . 5 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( dom `' ( f |` U. D ) i^i dom `' ( `' g |` ( A \ U. D ) ) ) = ( ( f " U. D ) i^i ( B \ ( f " U. D ) ) ) )
25		disjdif 3435	. . . . 5 |- ( ( f " U. D ) i^i ( B \ ( f " U. D ) ) ) = (/)
26	24, 25	syl6eq 2188	. . . 4 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( dom `' ( f |` U. D ) i^i dom `' ( `' g |` ( A \ U. D ) ) ) = (/) )
27	8, 10, 11, 12, 26	syl121anc 1221	. . 3 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ( dom `' ( f |` U. D ) i^i dom `' ( `' g |` ( A \ U. D ) ) ) = (/) )
28		funun 5167	. . 3 |- ( ( ( Fun `' ( f |` U. D ) /\ Fun `' ( `' g |` ( A \ U. D ) ) ) /\ ( dom `' ( f |` U. D ) i^i dom `' ( `' g |` ( A \ U. D ) ) ) = (/) ) -> Fun ( `' ( f |` U. D ) u. `' ( `' g |` ( A \ U. D ) ) ) )
29	2, 7, 27, 28	syl21anc 1215	. 2 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun ( `' ( f |` U. D ) u. `' ( `' g |` ( A \ U. D ) ) ) )
30		sbthlem.3	. . . . 5 |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
31	30	cnveqi 4714	. . . 4 |- `' H = `' ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
32		cnvun 4944	. . . 4 |- `' ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) ) = ( `' ( f |` U. D ) u. `' ( `' g |` ( A \ U. D ) ) )
33	31, 32	eqtri 2160	. . 3 |- `' H = ( `' ( f |` U. D ) u. `' ( `' g |` ( A \ U. D ) ) )
34	33	funeqi 5144	. 2 |- ( Fun `' H <-> Fun ( `' ( f |` U. D ) u. `' ( `' g |` ( A \ U. D ) ) ) )
35	29, 34	sylibr 133	1 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' H )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   {cab 2125   _Vcvv 2686   \ cdif 3068   u. cun 3069   i^i cin 3070   C_ wss 3071   (/)c0 3363   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