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Theorem rinvf1o 23038
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 . . . . . 6  |-  Fun  F
2 fdmrn 23035 . . . . . 6  |-  ( Fun 
F  <->  F : dom  F --> ran  F )
31, 2mpbi 199 . . . . 5  |-  F : dom  F --> ran  F
4 rinvbij.2 . . . . . . 7  |-  `' F  =  F
54funeqi 5275 . . . . . 6  |-  ( Fun  `' F  <->  Fun  F )
61, 5mpbir 200 . . . . 5  |-  Fun  `' F
73, 6pm3.2i 441 . . . 4  |-  ( F : dom  F --> ran  F  /\  Fun  `' F )
8 df-f1 5260 . . . 4  |-  ( F : dom  F -1-1-> ran  F  <-> 
( F : dom  F --> ran  F  /\  Fun  `' F ) )
97, 8mpbir 200 . . 3  |-  F : dom  F -1-1-> ran  F
10 rinvbij.4a . . 3  |-  A  C_  dom  F
11 f1ores 5487 . . 3  |-  ( ( F : dom  F -1-1-> ran 
F  /\  A  C_  dom  F )  ->  ( F  |`  A ) : A -1-1-onto-> ( F " A ) )
129, 10, 11mp2an 653 . 2  |-  ( F  |`  A ) : A -1-1-onto-> ( F " A )
13 rinvbij.3a . . . 4  |-  ( F
" A )  C_  B
14 rinvbij.3b . . . . . 6  |-  ( F
" B )  C_  A
15 rinvbij.4b . . . . . . 7  |-  B  C_  dom  F
16 funimass3 5641 . . . . . . 7  |-  ( ( Fun  F  /\  B  C_ 
dom  F )  -> 
( ( F " B )  C_  A  <->  B 
C_  ( `' F " A ) ) )
171, 15, 16mp2an 653 . . . . . 6  |-  ( ( F " B ) 
C_  A  <->  B  C_  ( `' F " A ) )
1814, 17mpbi 199 . . . . 5  |-  B  C_  ( `' F " A )
194imaeq1i 5009 . . . . . 6  |-  ( `' F " A )  =  ( F " A )
2019sseq2i 3203 . . . . 5  |-  ( B 
C_  ( `' F " A )  <->  B  C_  ( F " A ) )
2118, 20mpbi 199 . . . 4  |-  B  C_  ( F " A )
2213, 21eqssi 3195 . . 3  |-  ( F
" A )  =  B
23 f1oeq3 5465 . . 3  |-  ( ( F " A )  =  B  ->  (
( F  |`  A ) : A -1-1-onto-> ( F " A
)  <->  ( F  |`  A ) : A -1-1-onto-> B
) )
2422, 23ax-mp 8 . 2  |-  ( ( F  |`  A ) : A -1-1-onto-> ( F " A
)  <->  ( F  |`  A ) : A -1-1-onto-> B
)
2512, 24mpbi 199 1  |-  ( F  |`  A ) : A -1-1-onto-> B
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    C_ wss 3152   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249   -->wf 5251   -1-1->wf1 5252   -1-1-onto->wf1o 5254
This theorem is referenced by:  ballotlem7  23094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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