Theorem sbthlemi8 6673

Description: Lemma for isbth 6676. (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 5086	. . . 4 |- ( Fun `' f -> Fun `' ( f |` U. D ) )
2	1	ad2antlr 473	. . 3 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' ( f |` U. D ) )
3		funcnvcnv 5073	. . . . . 6 |- ( Fun g -> Fun `' `' g )
4		funres11 5086	. . . . . 6 |- ( Fun `' `' g -> Fun `' ( `' g |` ( A \ U. D ) ) )
5	3, 4	syl 14	. . . . 5 |- ( Fun g -> Fun `' ( `' g |` ( A \ U. D ) ) )
6	5	ad2antrr 472	. . . 4 |- ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) -> Fun `' ( `' g |` ( A \ U. D ) ) )
7	6	ad2antrl 474	. . 3 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' ( `' g |` ( A \ U. D ) ) )
8		simpll 496	. . . 4 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> EXMID )
9		simprll 504	. . . . 5 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ( Fun g /\ dom g = B ) )
10	9	simprd 112	. . . 4 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> dom g = B )
11		simprlr 505	. . . 4 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ran g C_ A )
12		simprr 499	. . . 4 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' g )
13		df-ima 4451	. . . . . . 7 |- ( f " U. D ) = ran ( f |` U. D )
14		df-rn 4449	. . . . . . 7 |- ran ( f |` U. D ) = dom `' ( f |` U. D )
15	13, 14	eqtr2i 2109	. . . . . 6 |- dom `' ( f |` U. D ) = ( f " U. D )
16		df-ima 4451	. . . . . . . 8 |- ( `' g " ( A \ U. D ) ) = ran ( `' g |` ( A \ U. D ) )
17		df-rn 4449	. . . . . . . 8 |- ran ( `' g |` ( A \ U. D ) ) = dom `' ( `' g |` ( A \ U. D ) )
18	16, 17	eqtri 2108	. . . . . . 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 6669	. . . . . . 7 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
22	18, 21	syl5eqr 2134	. . . . . 6 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> dom `' ( `' g |` ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
23		ineq12 3196	. . . . . 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 405	. . . . 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 3355	. . . . 5 |- ( ( f " U. D ) i^i ( B \ ( f " U. D ) ) ) = (/)
26	24, 25	syl6eq 2136	. . . 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 1179	. . 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 5058	. . 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 1173	. 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 4611	. . . 4 |- `' H = `' ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
32		cnvun 4837	. . . 4 |- `' ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) ) = ( `' ( f |` U. D ) u. `' ( `' g |` ( A \ U. D ) ) )
33	31, 32	eqtri 2108	. . 3 |- `' H = ( `' ( f |` U. D ) u. `' ( `' g |` ( A \ U. D ) ) )
34	33	funeqi 5036	. 2 |- ( Fun `' H <-> Fun ( `' ( f |` U. D ) u. `' ( `' g |` ( A \ U. D ) ) ) )
35	29, 34	sylibr 132	1 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' H )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 924   = wceq 1289   e. wcel 1438   {cab 2074   _Vcvv 2619   \ cdif 2996   u. cun 2997   i^i cin 2998   C_ wss 2999   (/)c0 3286   U.cuni 3653  EXMIDwem 4029   `'ccnv 4437   dom cdm 4438   ran crn 4439   |` cres 4440   "cima 4441   Fun wfun 5009

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-stab 776  df-dc 781  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-exmid 4030  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-fun 5017

This theorem is referenced by:  sbthlemi9  6674