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Theorem rinvf1o 27610
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 5654 . . . . 5  |-  ( Fun 
F  <->  F : dom  F --> ran  F )
31, 2mpbi 208 . . . 4  |-  F : dom  F --> ran  F
4 rinvbij.2 . . . . . 6  |-  `' F  =  F
54funeqi 5516 . . . . 5  |-  ( Fun  `' F  <->  Fun  F )
61, 5mpbir 209 . . . 4  |-  Fun  `' F
7 df-f1 5501 . . . 4  |-  ( F : dom  F -1-1-> ran  F  <-> 
( F : dom  F --> ran  F  /\  Fun  `' F ) )
83, 6, 7mpbir2an 918 . . 3  |-  F : dom  F -1-1-> ran  F
9 rinvbij.4a . . 3  |-  A  C_  dom  F
10 f1ores 5738 . . 3  |-  ( ( F : dom  F -1-1-> ran 
F  /\  A  C_  dom  F )  ->  ( F  |`  A ) : A -1-1-onto-> ( F " A ) )
118, 9, 10mp2an 670 . 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 5905 . . . . . . 7  |-  ( ( Fun  F  /\  B  C_ 
dom  F )  -> 
( ( F " B )  C_  A  <->  B 
C_  ( `' F " A ) ) )
161, 14, 15mp2an 670 . . . . . 6  |-  ( ( F " B ) 
C_  A  <->  B  C_  ( `' F " A ) )
1713, 16mpbi 208 . . . . 5  |-  B  C_  ( `' F " A )
184imaeq1i 5246 . . . . 5  |-  ( `' F " A )  =  ( F " A )
1917, 18sseqtri 3449 . . . 4  |-  B  C_  ( F " A )
2012, 19eqssi 3433 . . 3  |-  ( F
" A )  =  B
21 f1oeq3 5717 . . 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 1399    C_ wss 3389   `'ccnv 4912   dom cdm 4913   ran crn 4914    |` cres 4915   "cima 4916   Fun wfun 5490   -->wf 5492   -1-1->wf1 5493   -1-1-onto->wf1o 5495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504
This theorem is referenced by:  ballotlem7  28657
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