Theorem sbthlemi8 7206

Description: Lemma for isbth 7209. (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 5409	. . . 4 |- ( Fun `' f -> Fun `' ( f |` U. D ) )
2	1	ad2antlr 489	. . 3 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' ( f |` U. D ) )
3		funcnvcnv 5396	. . . . . 6 |- ( Fun g -> Fun `' `' g )
4		funres11 5409	. . . . . 6 |- ( Fun `' `' g -> Fun `' ( `' g |` ( A \ U. D ) ) )
5	3, 4	syl 14	. . . . 5 |- ( Fun g -> Fun `' ( `' g |` ( A \ U. D ) ) )
6	5	ad2antrr 488	. . . 4 |- ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) -> Fun `' ( `' g |` ( A \ U. D ) ) )
7	6	ad2antrl 490	. . 3 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' ( `' g |` ( A \ U. D ) ) )
8		simpll 527	. . . 4 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> EXMID )
9		simprll 539	. . . . 5 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ( Fun g /\ dom g = B ) )
10	9	simprd 114	. . . 4 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> dom g = B )
11		simprlr 540	. . . 4 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ran g C_ A )
12		simprr 533	. . . 4 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' g )
13		df-ima 4744	. . . . . . 7 |- ( f " U. D ) = ran ( f |` U. D )
14		df-rn 4742	. . . . . . 7 |- ran ( f |` U. D ) = dom `' ( f |` U. D )
15	13, 14	eqtr2i 2253	. . . . . 6 |- dom `' ( f |` U. D ) = ( f " U. D )
16		df-ima 4744	. . . . . . . 8 |- ( `' g " ( A \ U. D ) ) = ran ( `' g |` ( A \ U. D ) )
17		df-rn 4742	. . . . . . . 8 |- ran ( `' g |` ( A \ U. D ) ) = dom `' ( `' g |` ( A \ U. D ) )
18	16, 17	eqtri 2252	. . . . . . 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 7202	. . . . . . 7 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
22	18, 21	eqtr3id 2278	. . . . . 6 |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> dom `' ( `' g |` ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
23		ineq12 3405	. . . . . 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 414	. . . . 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 3569	. . . . 5 |- ( ( f " U. D ) i^i ( B \ ( f " U. D ) ) ) = (/)
26	24, 25	eqtrdi 2280	. . . 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 1279	. . 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 5378	. . 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 1273	. 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 4911	. . . 4 |- `' H = `' ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
32		cnvun 5149	. . . 4 |- `' ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) ) = ( `' ( f |` U. D ) u. `' ( `' g |` ( A \ U. D ) ) )
33	31, 32	eqtri 2252	. . 3 |- `' H = ( `' ( f |` U. D ) u. `' ( `' g |` ( A \ U. D ) ) )
34	33	funeqi 5354	. 2 |- ( Fun `' H <-> Fun ( `' ( f |` U. D ) u. `' ( `' g |` ( A \ U. D ) ) ) )
35	29, 34	sylibr 134	1 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' H )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2202   {cab 2217   _Vcvv 2803   \ cdif 3198   u. cun 3199   i^i cin 3200   C_ wss 3201   (/)c0 3496   U.cuni 3898  EXMIDwem 4290   `'ccnv 4730   dom cdm 4731   ran crn 4732   |` cres 4733   "cima 4734   Fun wfun 5327

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-exmid 4291  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-fun 5335

This theorem is referenced by:  sbthlemi9  7207