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Theorem rinvf1o 23054
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 23051 . . . . . 6  |-  ( Fun 
F  <->  F : dom  F --> ran  F )
31, 2mpbi 199 . . . . 5  |-  F : dom  F --> ran  F
4 rinvbij.2 . . . . . . 7  |-  `' F  =  F
54funeqi 5291 . . . . . 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 5276 . . . 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 5503 . . 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 5657 . . . . . . 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 5025 . . . . . 6  |-  ( `' F " A )  =  ( F " A )
2019sseq2i 3216 . . . . 5  |-  ( B 
C_  ( `' F " A )  <->  B  C_  ( F " A ) )
2118, 20mpbi 199 . . . 4  |-  B  C_  ( F " A )
2213, 21eqssi 3208 . . 3  |-  ( F
" A )  =  B
23 f1oeq3 5481 . . 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 1632    C_ wss 3165   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265   -->wf 5267   -1-1->wf1 5268   -1-1-onto->wf1o 5270
This theorem is referenced by:  ballotlem7  23110
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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