Theorem sbthlemi9 7093

Description: Lemma for isbth 7095. (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 5295	. . . . . . . . . 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 5294	. . . . . . . 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 5293	. . . . . 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 5295	. . . . . 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 7091	. . . 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 5294	. . . . . 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 7089	. . . 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 5293	. . 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 5293	. . . . . 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 7092	. . . 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 4704	. . . . 5 |- ran H = dom `' H
41	15, 16, 17	sbthlemi6 7090	. . . . 5 |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B )
42	40, 41	eqtr3id 2254	. . . 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 5293	. . 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 5552	. 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 2178   {cab 2193   _Vcvv 2776   \ cdif 3171   u. cun 3172   C_ wss 3174   U.cuni 3864  EXMIDwem 4254   `'ccnv 4692   dom cdm 4693   ran crn 4694   |` cres 4695   "cima 4696   Fun wfun 5284   Fn wfn 5285   -->wf 5286   -1-1->wf1 5287   -1-1-onto->wf1o 5289

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-exmid 4255  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297

This theorem is referenced by:  sbthlemi10  7094