Theorem sbthlem8 8077

Description: Lemma for sbth 8080. (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
sbthlem8	|- ( ( 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 sbthlem8

Step	Hyp	Ref	Expression
1		funres11 5966	. . . 4 |- ( Fun `' f -> Fun `' ( f |` U. D ) )
2		funcnvcnv 5956	. . . . . 6 |- ( Fun g -> Fun `' `' g )
3		funres11 5966	. . . . . 6 |- ( Fun `' `' g -> Fun `' ( `' g |` ( A \ U. D ) ) )
4	2, 3	syl 17	. . . . 5 |- ( Fun g -> Fun `' ( `' g |` ( A \ U. D ) ) )
5	4	ad3antrrr 766	. . . 4 |- ( ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) -> Fun `' ( `' g |` ( A \ U. D ) ) )
6	1, 5	anim12i 590	. . 3 |- ( ( Fun `' f /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ( Fun `' ( f |` U. D ) /\ Fun `' ( `' g |` ( A \ U. D ) ) ) )
7		df-ima 5127	. . . . . . . 8 |- ( f " U. D ) = ran ( f |` U. D )
8		df-rn 5125	. . . . . . . 8 |- ran ( f |` U. D ) = dom `' ( f |` U. D )
9	7, 8	eqtr2i 2645	. . . . . . 7 |- dom `' ( f |` U. D ) = ( f " U. D )
10		df-ima 5127	. . . . . . . . 9 |- ( `' g " ( A \ U. D ) ) = ran ( `' g |` ( A \ U. D ) )
11		df-rn 5125	. . . . . . . . 9 |- ran ( `' g |` ( A \ U. D ) ) = dom `' ( `' g |` ( A \ U. D ) )
12	10, 11	eqtri 2644	. . . . . . . 8 |- ( `' g " ( A \ U. D ) ) = dom `' ( `' g |` ( A \ U. D ) )
13		sbthlem.1	. . . . . . . . 9 |- A e. _V
14		sbthlem.2	. . . . . . . . 9 |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
15	13, 14	sbthlem4 8073	. . . . . . . 8 |- ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
16	12, 15	syl5eqr 2670	. . . . . . 7 |- ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> dom `' ( `' g |` ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
17		ineq12 3809	. . . . . . 7 |- ( ( 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 ) ) ) )
18	9, 16, 17	sylancr 695	. . . . . 6 |- ( ( ( 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 ) ) ) )
19		disjdif 4040	. . . . . 6 |- ( ( f " U. D ) i^i ( B \ ( f " U. D ) ) ) = (/)
20	18, 19	syl6eq 2672	. . . . 5 |- ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( dom `' ( f |` U. D ) i^i dom `' ( `' g |` ( A \ U. D ) ) ) = (/) )
21	20	adantlll 754	. . . 4 |- ( ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) -> ( dom `' ( f |` U. D ) i^i dom `' ( `' g |` ( A \ U. D ) ) ) = (/) )
22	21	adantl 482	. . 3 |- ( ( Fun `' f /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ( dom `' ( f |` U. D ) i^i dom `' ( `' g |` ( A \ U. D ) ) ) = (/) )
23		funun 5932	. . 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 ) ) ) )
24	6, 22, 23	syl2anc 693	. 2 |- ( ( Fun `' f /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun ( `' ( f |` U. D ) u. `' ( `' g |` ( A \ U. D ) ) ) )
25		sbthlem.3	. . . . 5 |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
26	25	cnveqi 5297	. . . 4 |- `' H = `' ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
27		cnvun 5538	. . . 4 |- `' ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) ) = ( `' ( f |` U. D ) u. `' ( `' g |` ( A \ U. D ) ) )
28	26, 27	eqtri 2644	. . 3 |- `' H = ( `' ( f |` U. D ) u. `' ( `' g |` ( A \ U. D ) ) )
29	28	funeqi 5909	. 2 |- ( Fun `' H <-> Fun ( `' ( f |` U. D ) u. `' ( `' g |` ( A \ U. D ) ) ) )
30	24, 29	sylibr 224	1 |- ( ( Fun `' f /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' H )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   U.cuni 4436   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890

This theorem is referenced by:  sbthlem9  8078