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

Theorem rinvf1o 24042
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 24039 . . . . 5  |-  ( Fun 
F  <->  F : dom  F --> ran  F )
31, 2mpbi 200 . . . 4  |-  F : dom  F --> ran  F
4 rinvbij.2 . . . . . 6  |-  `' F  =  F
54funeqi 5474 . . . . 5  |-  ( Fun  `' F  <->  Fun  F )
61, 5mpbir 201 . . . 4  |-  Fun  `' F
7 df-f1 5459 . . . 4  |-  ( F : dom  F -1-1-> ran  F  <-> 
( F : dom  F --> ran  F  /\  Fun  `' F ) )
83, 6, 7mpbir2an 887 . . 3  |-  F : dom  F -1-1-> ran  F
9 rinvbij.4a . . 3  |-  A  C_  dom  F
10 f1ores 5689 . . 3  |-  ( ( F : dom  F -1-1-> ran 
F  /\  A  C_  dom  F )  ->  ( F  |`  A ) : A -1-1-onto-> ( F " A ) )
118, 9, 10mp2an 654 . 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 5846 . . . . . . 7  |-  ( ( Fun  F  /\  B  C_ 
dom  F )  -> 
( ( F " B )  C_  A  <->  B 
C_  ( `' F " A ) ) )
161, 14, 15mp2an 654 . . . . . 6  |-  ( ( F " B ) 
C_  A  <->  B  C_  ( `' F " A ) )
1713, 16mpbi 200 . . . . 5  |-  B  C_  ( `' F " A )
184imaeq1i 5200 . . . . 5  |-  ( `' F " A )  =  ( F " A )
1917, 18sseqtri 3380 . . . 4  |-  B  C_  ( F " A )
2012, 19eqssi 3364 . . 3  |-  ( F
" A )  =  B
21 f1oeq3 5667 . . 3  |-  ( ( F " A )  =  B  ->  (
( F  |`  A ) : A -1-1-onto-> ( F " A
)  <->  ( F  |`  A ) : A -1-1-onto-> B
) )
2220, 21ax-mp 8 . 2  |-  ( ( F  |`  A ) : A -1-1-onto-> ( F " A
)  <->  ( F  |`  A ) : A -1-1-onto-> B
)
2311, 22mpbi 200 1  |-  ( F  |`  A ) : A -1-1-onto-> B
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    C_ wss 3320   `'ccnv 4877   dom cdm 4878   ran crn 4879    |` cres 4880   "cima 4881   Fun wfun 5448   -->wf 5450   -1-1->wf1 5451   -1-1-onto->wf1o 5453
This theorem is referenced by:  ballotlem7  24793
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462
  Copyright terms: Public domain W3C validator