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Theorem sbthlem9 7217
Description: Lemma for sbth 7219. (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
sbthlem9  |-  ( ( 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 sbthlem9
StepHypRef Expression
1 sbthlem.1 . . . . . . . 8  |-  A  e. 
_V
2 sbthlem.2 . . . . . . . 8  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
3 sbthlem.3 . . . . . . . 8  |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
\  U. D ) ) )
41, 2, 3sbthlem7 7215 . . . . . . 7  |-  ( ( Fun  f  /\  Fun  `' g )  ->  Fun  H )
51, 2, 3sbthlem5 7213 . . . . . . . 8  |-  ( ( dom  f  =  A  /\  ran  g  C_  A )  ->  dom  H  =  A )
65adantrl 697 . . . . . . 7  |-  ( ( dom  f  =  A  /\  ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A ) )  ->  dom  H  =  A )
74, 6anim12i 550 . . . . . 6  |-  ( ( ( Fun  f  /\  Fun  `' g )  /\  ( dom  f  =  A  /\  ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A ) ) )  ->  ( Fun  H  /\  dom  H  =  A ) )
87an42s 801 . . . . 5  |-  ( ( ( Fun  f  /\  dom  f  =  A
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  -> 
( Fun  H  /\  dom  H  =  A ) )
98adantlr 696 . . . 4  |-  ( ( ( ( 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 ) )
109adantlr 696 . . 3  |-  ( ( ( ( ( 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 ) )
111, 2, 3sbthlem8 7216 . . . 4  |-  ( ( Fun  `' f  /\  ( ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  `' H
)
1211adantll 695 . . 3  |-  ( ( ( ( ( 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 448 . . . . . . 7  |-  ( ( Fun  g  /\  dom  g  =  B )  ->  dom  g  =  B )
1413anim1i 552 . . . . . 6  |-  ( ( ( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  ->  ( dom  g  =  B  /\  ran  g  C_  A
) )
15 df-rn 4881 . . . . . . 7  |-  ran  H  =  dom  `' H
161, 2, 3sbthlem6 7214 . . . . . . 7  |-  ( ( ran  f  C_  B  /\  ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  ran  H  =  B )
1715, 16syl5eqr 2481 . . . . . 6  |-  ( ( ran  f  C_  B  /\  ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  dom  `' H  =  B
)
1814, 17sylanr1 634 . . . . 5  |-  ( ( ran  f  C_  B  /\  ( ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  dom  `' H  =  B )
1918adantll 695 . . . 4  |-  ( ( ( ( Fun  f  /\  dom  f  =  A )  /\  ran  f  C_  B )  /\  (
( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  dom  `' H  =  B
)
2019adantlr 696 . . 3  |-  ( ( ( ( ( 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 )
2110, 12, 20jca32 522 . 2  |-  ( ( ( ( ( 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 5451 . . . 4  |-  ( f : A -1-1-> B  <->  ( f : A --> B  /\  Fun  `' f ) )
23 df-f 5450 . . . . . 6  |-  ( f : A --> B  <->  ( f  Fn  A  /\  ran  f  C_  B ) )
24 df-fn 5449 . . . . . . 7  |-  ( f  Fn  A  <->  ( Fun  f  /\  dom  f  =  A ) )
2524anbi1i 677 . . . . . 6  |-  ( ( f  Fn  A  /\  ran  f  C_  B )  <-> 
( ( Fun  f  /\  dom  f  =  A )  /\  ran  f  C_  B ) )
2623, 25bitri 241 . . . . 5  |-  ( f : A --> B  <->  ( ( Fun  f  /\  dom  f  =  A )  /\  ran  f  C_  B ) )
2726anbi1i 677 . . . 4  |-  ( ( f : A --> B  /\  Fun  `' f )  <->  ( (
( Fun  f  /\  dom  f  =  A
)  /\  ran  f  C_  B )  /\  Fun  `' f ) )
2822, 27bitri 241 . . 3  |-  ( f : A -1-1-> B  <->  ( (
( Fun  f  /\  dom  f  =  A
)  /\  ran  f  C_  B )  /\  Fun  `' f ) )
29 df-f1 5451 . . . 4  |-  ( g : B -1-1-> A  <->  ( g : B --> A  /\  Fun  `' g ) )
30 df-f 5450 . . . . . 6  |-  ( g : B --> A  <->  ( g  Fn  B  /\  ran  g  C_  A ) )
31 df-fn 5449 . . . . . . 7  |-  ( g  Fn  B  <->  ( Fun  g  /\  dom  g  =  B ) )
3231anbi1i 677 . . . . . 6  |-  ( ( g  Fn  B  /\  ran  g  C_  A )  <-> 
( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A ) )
3330, 32bitri 241 . . . . 5  |-  ( g : B --> A  <->  ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A ) )
3433anbi1i 677 . . . 4  |-  ( ( g : B --> A  /\  Fun  `' g )  <->  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )
3529, 34bitri 241 . . 3  |-  ( g : B -1-1-> A  <->  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )
3628, 35anbi12i 679 . 2  |-  ( ( 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 5674 . . 3  |-  ( H : A -1-1-onto-> B  <->  ( H  Fn  A  /\  `' H  Fn  B ) )
38 df-fn 5449 . . . 4  |-  ( H  Fn  A  <->  ( Fun  H  /\  dom  H  =  A ) )
39 df-fn 5449 . . . 4  |-  ( `' H  Fn  B  <->  ( Fun  `' H  /\  dom  `' H  =  B )
)
4038, 39anbi12i 679 . . 3  |-  ( ( H  Fn  A  /\  `' H  Fn  B
)  <->  ( ( Fun 
H  /\  dom  H  =  A )  /\  ( Fun  `' H  /\  dom  `' H  =  B )
) )
4137, 40bitri 241 . 2  |-  ( H : A -1-1-onto-> B  <->  ( ( Fun 
H  /\  dom  H  =  A )  /\  ( Fun  `' H  /\  dom  `' H  =  B )
) )
4221, 36, 413imtr4i 258 1  |-  ( ( f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  H : A
-1-1-onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   _Vcvv 2948    \ cdif 3309    u. cun 3310    C_ wss 3312   U.cuni 4007   `'ccnv 4869   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   Fun wfun 5440    Fn wfn 5441   -->wf 5442   -1-1->wf1 5443   -1-1-onto->wf1o 5445
This theorem is referenced by:  sbthlem10  7218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453
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