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Theorem rinvf1o 27133
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 5739 . . . . 5  |-  ( Fun 
F  <->  F : dom  F --> ran  F )
31, 2mpbi 208 . . . 4  |-  F : dom  F --> ran  F
4 rinvbij.2 . . . . . 6  |-  `' F  =  F
54funeqi 5601 . . . . 5  |-  ( Fun  `' F  <->  Fun  F )
61, 5mpbir 209 . . . 4  |-  Fun  `' F
7 df-f1 5586 . . . 4  |-  ( F : dom  F -1-1-> ran  F  <-> 
( F : dom  F --> ran  F  /\  Fun  `' F ) )
83, 6, 7mpbir2an 913 . . 3  |-  F : dom  F -1-1-> ran  F
9 rinvbij.4a . . 3  |-  A  C_  dom  F
10 f1ores 5823 . . 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 5990 . . . . . . 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 5327 . . . . 5  |-  ( `' F " A )  =  ( F " A )
1917, 18sseqtri 3531 . . . 4  |-  B  C_  ( F " A )
2012, 19eqssi 3515 . . 3  |-  ( F
" A )  =  B
21 f1oeq3 5802 . . 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 1374    C_ wss 3471   `'ccnv 4993   dom cdm 4994   ran crn 4995    |` cres 4996   "cima 4997   Fun wfun 5575   -->wf 5577   -1-1->wf1 5578   -1-1-onto->wf1o 5580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589
This theorem is referenced by:  ballotlem7  28102
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