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Theorem sbthlem9 7161
Description: Lemma for sbth 7163. (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 7159 . . . . . . 7  |-  ( ( Fun  f  /\  Fun  `' g )  ->  Fun  H )
51, 2, 3sbthlem5 7157 . . . . . . . 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 7160 . . . 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 4829 . . . . . . 7  |-  ran  H  =  dom  `' H
161, 2, 3sbthlem6 7158 . . . . . . 7  |-  ( ( ran  f  C_  B  /\  ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  ran  H  =  B )
1715, 16syl5eqr 2433 . . . . . 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 5399 . . . 4  |-  ( f : A -1-1-> B  <->  ( f : A --> B  /\  Fun  `' f ) )
23 df-f 5398 . . . . . 6  |-  ( f : A --> B  <->  ( f  Fn  A  /\  ran  f  C_  B ) )
24 df-fn 5397 . . . . . . 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 5399 . . . 4  |-  ( g : B -1-1-> A  <->  ( g : B --> A  /\  Fun  `' g ) )
30 df-f 5398 . . . . . 6  |-  ( g : B --> A  <->  ( g  Fn  B  /\  ran  g  C_  A ) )
31 df-fn 5397 . . . . . . 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 5622 . . 3  |-  ( H : A -1-1-onto-> B  <->  ( H  Fn  A  /\  `' H  Fn  B ) )
38 df-fn 5397 . . . 4  |-  ( H  Fn  A  <->  ( Fun  H  /\  dom  H  =  A ) )
39 df-fn 5397 . . . 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 1649    e. wcel 1717   {cab 2373   _Vcvv 2899    \ cdif 3260    u. cun 3261    C_ wss 3263   U.cuni 3957   `'ccnv 4817   dom cdm 4818   ran crn 4819    |` cres 4820   "cima 4821   Fun wfun 5388    Fn wfn 5389   -->wf 5390   -1-1->wf1 5391   -1-1-onto->wf1o 5393
This theorem is referenced by:  sbthlem10  7162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401
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