Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rinvf1o Structured version   Unicode version

Theorem rinvf1o 25948
Description: Sufficient conditions for the restriction of an involution to be a bijection (Contributed by Thierry Arnoux, 7-Dec-2016.)
Hypotheses
Ref Expression
rinvbij.1  |-  Fun  F
rinvbij.2  |-  `' F  =  F
rinvbij.3a  |-  ( F
" A )  C_  B
rinvbij.3b  |-  ( F
" B )  C_  A
rinvbij.4a  |-  A  C_  dom  F
rinvbij.4b  |-  B  C_  dom  F
Assertion
Ref Expression
rinvf1o  |-  ( F  |`  A ) : A -1-1-onto-> B

Proof of Theorem rinvf1o
StepHypRef Expression
1 rinvbij.1 . . . . 5  |-  Fun  F
2 fdmrn 5572 . . . . 5  |-  ( Fun 
F  <->  F : dom  F --> ran  F )
31, 2mpbi 208 . . . 4  |-  F : dom  F --> ran  F
4 rinvbij.2 . . . . . 6  |-  `' F  =  F
54funeqi 5437 . . . . 5  |-  ( Fun  `' F  <->  Fun  F )
61, 5mpbir 209 . . . 4  |-  Fun  `' F
7 df-f1 5422 . . . 4  |-  ( F : dom  F -1-1-> ran  F  <-> 
( F : dom  F --> ran  F  /\  Fun  `' F ) )
83, 6, 7mpbir2an 911 . . 3  |-  F : dom  F -1-1-> ran  F
9 rinvbij.4a . . 3  |-  A  C_  dom  F
10 f1ores 5654 . . 3  |-  ( ( F : dom  F -1-1-> ran 
F  /\  A  C_  dom  F )  ->  ( F  |`  A ) : A -1-1-onto-> ( F " A ) )
118, 9, 10mp2an 672 . 2  |-  ( F  |`  A ) : A -1-1-onto-> ( F " A )
12 rinvbij.3a . . . 4  |-  ( F
" A )  C_  B
13 rinvbij.3b . . . . . 6  |-  ( F
" B )  C_  A
14 rinvbij.4b . . . . . . 7  |-  B  C_  dom  F
15 funimass3 5818 . . . . . . 7  |-  ( ( Fun  F  /\  B  C_ 
dom  F )  -> 
( ( F " B )  C_  A  <->  B 
C_  ( `' F " A ) ) )
161, 14, 15mp2an 672 . . . . . 6  |-  ( ( F " B ) 
C_  A  <->  B  C_  ( `' F " A ) )
1713, 16mpbi 208 . . . . 5  |-  B  C_  ( `' F " A )
184imaeq1i 5165 . . . . 5  |-  ( `' F " A )  =  ( F " A )
1917, 18sseqtri 3387 . . . 4  |-  B  C_  ( F " A )
2012, 19eqssi 3371 . . 3  |-  ( F
" A )  =  B
21 f1oeq3 5633 . . 3  |-  ( ( F " A )  =  B  ->  (
( F  |`  A ) : A -1-1-onto-> ( F " A
)  <->  ( F  |`  A ) : A -1-1-onto-> B
) )
2220, 21ax-mp 5 . 2  |-  ( ( F  |`  A ) : A -1-1-onto-> ( F " A
)  <->  ( F  |`  A ) : A -1-1-onto-> B
)
2311, 22mpbi 208 1  |-  ( F  |`  A ) : A -1-1-onto-> B
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369    C_ wss 3327   `'ccnv 4838   dom cdm 4839   ran crn 4840    |` cres 4841   "cima 4842   Fun wfun 5411   -->wf 5413   -1-1->wf1 5414   -1-1-onto->wf1o 5416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425
This theorem is referenced by:  ballotlem7  26917
  Copyright terms: Public domain W3C validator