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Theorem sbthlem9 7637
Description: Lemma for sbth 7639. (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 7635 . . . . . . 7  |-  ( ( Fun  f  /\  Fun  `' g )  ->  Fun  H )
51, 2, 3sbthlem5 7633 . . . . . . . 8  |-  ( ( dom  f  =  A  /\  ran  g  C_  A )  ->  dom  H  =  A )
65adantrl 715 . . . . . . 7  |-  ( ( dom  f  =  A  /\  ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A ) )  ->  dom  H  =  A )
74, 6anim12i 566 . . . . . 6  |-  ( ( ( Fun  f  /\  Fun  `' g )  /\  ( dom  f  =  A  /\  ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A ) ) )  ->  ( Fun  H  /\  dom  H  =  A ) )
87an42s 827 . . . . 5  |-  ( ( ( Fun  f  /\  dom  f  =  A
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  -> 
( Fun  H  /\  dom  H  =  A ) )
98adantlr 714 . . . 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 714 . . 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 7636 . . . 4  |-  ( ( Fun  `' f  /\  ( ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  `' H
)
1211adantll 713 . . 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 461 . . . . . . 7  |-  ( ( Fun  g  /\  dom  g  =  B )  ->  dom  g  =  B )
1413anim1i 568 . . . . . 6  |-  ( ( ( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  ->  ( dom  g  =  B  /\  ran  g  C_  A
) )
15 df-rn 5000 . . . . . . 7  |-  ran  H  =  dom  `' H
161, 2, 3sbthlem6 7634 . . . . . . 7  |-  ( ( ran  f  C_  B  /\  ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  ran  H  =  B )
1715, 16syl5eqr 2498 . . . . . 6  |-  ( ( ran  f  C_  B  /\  ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  dom  `' H  =  B
)
1814, 17sylanr1 652 . . . . 5  |-  ( ( ran  f  C_  B  /\  ( ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  dom  `' H  =  B )
1918adantll 713 . . . 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 714 . . 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 535 . 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 5583 . . . 4  |-  ( f : A -1-1-> B  <->  ( f : A --> B  /\  Fun  `' f ) )
23 df-f 5582 . . . . . 6  |-  ( f : A --> B  <->  ( f  Fn  A  /\  ran  f  C_  B ) )
24 df-fn 5581 . . . . . . 7  |-  ( f  Fn  A  <->  ( Fun  f  /\  dom  f  =  A ) )
2524anbi1i 695 . . . . . 6  |-  ( ( f  Fn  A  /\  ran  f  C_  B )  <-> 
( ( Fun  f  /\  dom  f  =  A )  /\  ran  f  C_  B ) )
2623, 25bitri 249 . . . . 5  |-  ( f : A --> B  <->  ( ( Fun  f  /\  dom  f  =  A )  /\  ran  f  C_  B ) )
2726anbi1i 695 . . . 4  |-  ( ( f : A --> B  /\  Fun  `' f )  <->  ( (
( Fun  f  /\  dom  f  =  A
)  /\  ran  f  C_  B )  /\  Fun  `' f ) )
2822, 27bitri 249 . . 3  |-  ( f : A -1-1-> B  <->  ( (
( Fun  f  /\  dom  f  =  A
)  /\  ran  f  C_  B )  /\  Fun  `' f ) )
29 df-f1 5583 . . . 4  |-  ( g : B -1-1-> A  <->  ( g : B --> A  /\  Fun  `' g ) )
30 df-f 5582 . . . . . 6  |-  ( g : B --> A  <->  ( g  Fn  B  /\  ran  g  C_  A ) )
31 df-fn 5581 . . . . . . 7  |-  ( g  Fn  B  <->  ( Fun  g  /\  dom  g  =  B ) )
3231anbi1i 695 . . . . . 6  |-  ( ( g  Fn  B  /\  ran  g  C_  A )  <-> 
( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A ) )
3330, 32bitri 249 . . . . 5  |-  ( g : B --> A  <->  ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A ) )
3433anbi1i 695 . . . 4  |-  ( ( g : B --> A  /\  Fun  `' g )  <->  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )
3529, 34bitri 249 . . 3  |-  ( g : B -1-1-> A  <->  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )
3628, 35anbi12i 697 . 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 5814 . . 3  |-  ( H : A -1-1-onto-> B  <->  ( H  Fn  A  /\  `' H  Fn  B ) )
38 df-fn 5581 . . . 4  |-  ( H  Fn  A  <->  ( Fun  H  /\  dom  H  =  A ) )
39 df-fn 5581 . . . 4  |-  ( `' H  Fn  B  <->  ( Fun  `' H  /\  dom  `' H  =  B )
)
4038, 39anbi12i 697 . . 3  |-  ( ( H  Fn  A  /\  `' H  Fn  B
)  <->  ( ( Fun 
H  /\  dom  H  =  A )  /\  ( Fun  `' H  /\  dom  `' H  =  B )
) )
4137, 40bitri 249 . 2  |-  ( H : A -1-1-onto-> B  <->  ( ( Fun 
H  /\  dom  H  =  A )  /\  ( Fun  `' H  /\  dom  `' H  =  B )
) )
4221, 36, 413imtr4i 266 1  |-  ( ( f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  H : A
-1-1-onto-> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   {cab 2428   _Vcvv 3095    \ cdif 3458    u. cun 3459    C_ wss 3461   U.cuni 4234   `'ccnv 4988   dom cdm 4989   ran crn 4990    |` cres 4991   "cima 4992   Fun wfun 5572    Fn wfn 5573   -->wf 5574   -1-1->wf1 5575   -1-1-onto->wf1o 5577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585
This theorem is referenced by:  sbthlem10  7638
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