Metamath Proof Explorer

Theorem sbthlem9

Description: Lemma for sbth . (Contributed by NM, 28-Mar-1998)

Ref Expression
Hypotheses 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 sbthlem9 
|- ( ( f : A -1-1-> B /\ g : B -1-1-> A ) -> H : A -1-1-onto-> B )

Proof

Step Hyp Ref Expression
1 sbthlem.1 
 |-  A e. _V
2 sbthlem.2 
 |-  D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
3 sbthlem.3 
 |-  H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
4 1 2 3 sbthlem7 
 |-  ( ( Fun f /\ Fun `' g ) -> Fun H )
5 1 2 3 sbthlem5 
 |-  ( ( dom f = A /\ ran g C_ A ) -> dom H = A )
6 5 adantrl 
 |-  ( ( dom f = A /\ ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) ) -> dom H = A )
7 4 6 anim12i 
 |-  ( ( ( Fun f /\ Fun `' g ) /\ ( dom f = A /\ ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) ) ) -> ( Fun H /\ dom H = A ) )
8 7 an42s 
 |-  ( ( ( Fun f /\ dom f = A ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ( Fun H /\ dom H = A ) )
9 8 adantlr 
 |-  ( ( ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ( Fun H /\ dom H = A ) )
10 9 adantlr 
 |-  ( ( ( ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ( Fun H /\ dom H = A ) )
11 1 2 3 sbthlem8 
 |-  ( ( Fun `' f /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' H )
12 11 adantll 
 |-  ( ( ( ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' H )
13 simpr 
 |-  ( ( Fun g /\ dom g = B ) -> dom g = B )
14 13 anim1i 
 |-  ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) -> ( dom g = B /\ ran g C_ A ) )
15 df-rn 
 |-  ran H = dom `' H
16 1 2 3 sbthlem6 
 |-  ( ( ran f C_ B /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B )
17 15 16 eqtr3id 
 |-  ( ( ran f C_ B /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> dom `' H = B )
18 14 17 sylanr1 
 |-  ( ( ran f C_ B /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> dom `' H = B )
19 18 adantll 
 |-  ( ( ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> dom `' H = B )
20 19 adantlr 
 |-  ( ( ( ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> dom `' H = B )
21 10 12 20 jca32 
 |-  ( ( ( ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> ( ( Fun H /\ dom H = A ) /\ ( Fun `' H /\ dom `' H = B ) ) )
22 df-f1 
 |-  ( f : A -1-1-> B <-> ( f : A --> B /\ Fun `' f ) )
23 df-f 
 |-  ( f : A --> B <-> ( f Fn A /\ ran f C_ B ) )
24 df-fn 
 |-  ( f Fn A <-> ( Fun f /\ dom f = A ) )
25 24 anbi1i 
 |-  ( ( f Fn A /\ ran f C_ B ) <-> ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) )
26 23 25 bitri 
 |-  ( f : A --> B <-> ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) )
27 26 anbi1i 
 |-  ( ( f : A --> B /\ Fun `' f ) <-> ( ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) /\ Fun `' f ) )
28 22 27 bitri 
 |-  ( f : A -1-1-> B <-> ( ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) /\ Fun `' f ) )
29 df-f1 
 |-  ( g : B -1-1-> A <-> ( g : B --> A /\ Fun `' g ) )
30 df-f 
 |-  ( g : B --> A <-> ( g Fn B /\ ran g C_ A ) )
31 df-fn 
 |-  ( g Fn B <-> ( Fun g /\ dom g = B ) )
32 31 anbi1i 
 |-  ( ( g Fn B /\ ran g C_ A ) <-> ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) )
33 30 32 bitri 
 |-  ( g : B --> A <-> ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) )
34 33 anbi1i 
 |-  ( ( g : B --> A /\ Fun `' g ) <-> ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) )
35 29 34 bitri 
 |-  ( g : B -1-1-> A <-> ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) )
36 28 35 anbi12i 
 |-  ( ( f : A -1-1-> B /\ g : B -1-1-> A ) <-> ( ( ( ( Fun f /\ dom f = A ) /\ ran f C_ B ) /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) )
37 dff1o4 
 |-  ( H : A -1-1-onto-> B <-> ( H Fn A /\ `' H Fn B ) )
38 df-fn 
 |-  ( H Fn A <-> ( Fun H /\ dom H = A ) )
39 df-fn 
 |-  ( `' H Fn B <-> ( Fun `' H /\ dom `' H = B ) )
40 38 39 anbi12i 
 |-  ( ( H Fn A /\ `' H Fn B ) <-> ( ( Fun H /\ dom H = A ) /\ ( Fun `' H /\ dom `' H = B ) ) )
41 37 40 bitri 
 |-  ( H : A -1-1-onto-> B <-> ( ( Fun H /\ dom H = A ) /\ ( Fun `' H /\ dom `' H = B ) ) )
42 21 36 41 3imtr4i 
 |-  ( ( f : A -1-1-> B /\ g : B -1-1-> A ) -> H : A -1-1-onto-> B )