Theorem sbthlemi9 7069

Description: Lemma for isbth 7071. (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 1001	. . . . . . . . . 10 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> f : A -1-1-> B )
2		df-f1 5277	. . . . . . . . . 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 5276	. . . . . . . 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 5275	. . . . . 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 1002	. . . . . 6 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> g : B -1-1-> A )
12		df-f1 5277	. . . . . 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 7067	. . . 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 1000	. . . 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 5276	. . . . . 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 7065	. . . 4 |- ( (EXMID /\ ( dom f = A /\ ran g C_ A ) ) -> dom H = A )
27	20, 21, 25, 26	syl12anc 1248	. . 3 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> dom H = A )
28		df-fn 5275	. . 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 5275	. . . . . 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 7068	. . . 4 |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' H )
36	20, 30, 34, 14, 35	syl22anc 1251	. . 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 4687	. . . . 5 |- ran H = dom `' H
41	15, 16, 17	sbthlemi6 7066	. . . . 5 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B )
42	40, 41	eqtr3id 2252	. . . 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 1251	. . 3 |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> dom `' H = B )
44		df-fn 5275	. . 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 5532	. 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 981   = wceq 1373   e. wcel 2176   {cab 2191   _Vcvv 2772   \ cdif 3163   u. cun 3164   C_ wss 3166   U.cuni 3850  EXMIDwem 4239   `'ccnv 4675   dom cdm 4676   ran crn 4677   |` cres 4678   "cima 4679   Fun wfun 5266   Fn wfn 5267   -->wf 5268   -1-1->wf1 5269   -1-1-onto->wf1o 5271

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-exmid 4240  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279

This theorem is referenced by:  sbthlemi10  7070