Theorem sbthlemi9 7132

Description: Lemma for isbth 7134. (Contributed by NM, 28-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
sbthlemi9	|- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> H : A -1-1-onto-> 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 sbthlemi9

Step	Hyp	Ref	Expression
1		simp2 1022	. . . . . . . . . 10 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> f : A -1-1-> B )
2		df-f1 5323	. . . . . . . . . 10 |- ( f : A -1-1-> B <-> ( f : A --> B /\ Fun `' f ) )
3	1, 2	sylib 122	. . . . . . . . 9 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> ( f : A --> B /\ Fun `' f ) )
4	3	simpld 112	. . . . . . . 8 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> f : A --> B )
5		df-f 5322	. . . . . . . 8 |- ( f : A --> B <-> ( f Fn A /\ ran f C_ B ) )
6	4, 5	sylib 122	. . . . . . 7 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> ( f Fn A /\ ran f C_ B ) )
7	6	simpld 112	. . . . . 6 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> f Fn A )
8		df-fn 5321	. . . . . 6 |- ( f Fn A <-> ( Fun f /\ dom f = A ) )
9	7, 8	sylib 122	. . . . 5 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> ( Fun f /\ dom f = A ) )
10	9	simpld 112	. . . 4 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> Fun f )
11		simp3 1023	. . . . . 6 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> g : B -1-1-> A )
12		df-f1 5323	. . . . . 6 |- ( g : B -1-1-> A <-> ( g : B --> A /\ Fun `' g ) )
13	11, 12	sylib 122	. . . . 5 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> ( g : B --> A /\ Fun `' g ) )
14	13	simprd 114	. . . 4 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> Fun `' g )
15		sbthlem.1	. . . . 5 |- A e. _V
16		sbthlem.2	. . . . 5 |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
17		sbthlem.3	. . . . 5 |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
18	15, 16, 17	sbthlem7 7130	. . . 4 |- ( ( Fun f /\ Fun `' g ) -> Fun H )
19	10, 14, 18	syl2anc 411	. . 3 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> Fun H )
20		simp1 1021	. . . 4 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> EXMID )
21	9	simprd 114	. . . 4 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> dom f = A )
22	13	simpld 112	. . . . . 6 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> g : B --> A )
23		df-f 5322	. . . . . 6 |- ( g : B --> A <-> ( g Fn B /\ ran g C_ A ) )
24	22, 23	sylib 122	. . . . 5 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> ( g Fn B /\ ran g C_ A ) )
25	24	simprd 114	. . . 4 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> ran g C_ A )
26	15, 16, 17	sbthlemi5 7128	. . . 4 |- ( (EXMID /\ ( dom f = A /\ ran g C_ A ) ) -> dom H = A )
27	20, 21, 25, 26	syl12anc 1269	. . 3 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> dom H = A )
28		df-fn 5321	. . 3 |- ( H Fn A <-> ( Fun H /\ dom H = A ) )
29	19, 27, 28	sylanbrc 417	. 2 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> H Fn A )
30	3	simprd 114	. . . 4 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> Fun `' f )
31	24	simpld 112	. . . . . 6 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> g Fn B )
32		df-fn 5321	. . . . . 6 |- ( g Fn B <-> ( Fun g /\ dom g = B ) )
33	31, 32	sylib 122	. . . . 5 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> ( Fun g /\ dom g = B ) )
34	33, 25	jca 306	. . . 4 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) )
35	15, 16, 17	sbthlemi8 7131	. . . 4 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' H )
36	20, 30, 34, 14, 35	syl22anc 1272	. . 3 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> Fun `' H )
37	6	simprd 114	. . . 4 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> ran f C_ B )
38	33	simprd 114	. . . . 5 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> dom g = B )
39	38, 25	jca 306	. . . 4 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> ( dom g = B /\ ran g C_ A ) )
40		df-rn 4730	. . . . 5 |- ran H = dom `' H
41	15, 16, 17	sbthlemi6 7129	. . . . 5 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B )
42	40, 41	eqtr3id 2276	. . . 4 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> dom `' H = B )
43	20, 37, 39, 14, 42	syl22anc 1272	. . 3 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> dom `' H = B )
44		df-fn 5321	. . 3 |- ( `' H Fn B <-> ( Fun `' H /\ dom `' H = B ) )
45	36, 43, 44	sylanbrc 417	. 2 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> `' H Fn B )
46		dff1o4 5580	. 2 |- ( H : A -1-1-onto-> B <-> ( H Fn A /\ `' H Fn B ) )
47	29, 45, 46	sylanbrc 417	1 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> H : A -1-1-onto-> B )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   {cab 2215   _Vcvv 2799   \ cdif 3194   u. cun 3195   C_ wss 3197   U.cuni 3888  EXMIDwem 4278   `'ccnv 4718   dom cdm 4719   ran crn 4720   |` cres 4721   "cima 4722   Fun wfun 5312   Fn wfn 5313   -->wf 5314   -1-1->wf1 5315   -1-1-onto->wf1o 5317

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-exmid 4279  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325

This theorem is referenced by:  sbthlemi10  7133