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Theorem f1opwfi 7896
Description: A one-to-one mapping induces a one-to-one mapping on finite subsets. (Contributed by Mario Carneiro, 25-Jan-2015.)
Assertion
Ref Expression
f1opwfi  |-  ( F : A -1-1-onto-> B  ->  ( b  e.  ( ~P A  i^i  Fin )  |->  ( F "
b ) ) : ( ~P A  i^i  Fin ) -1-1-onto-> ( ~P B  i^i  Fin ) )
Distinct variable groups:    A, b    B, b    F, b

Proof of Theorem f1opwfi
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 eqid 2471 . 2  |-  ( b  e.  ( ~P A  i^i  Fin )  |->  ( F
" b ) )  =  ( b  e.  ( ~P A  i^i  Fin )  |->  ( F "
b ) )
2 imassrn 5185 . . . . . 6  |-  ( F
" b )  C_  ran  F
3 f1ofo 5835 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -onto-> B )
4 forn 5809 . . . . . . 7  |-  ( F : A -onto-> B  ->  ran  F  =  B )
53, 4syl 17 . . . . . 6  |-  ( F : A -1-1-onto-> B  ->  ran  F  =  B )
62, 5syl5sseq 3466 . . . . 5  |-  ( F : A -1-1-onto-> B  ->  ( F " b )  C_  B
)
76adantr 472 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F "
b )  C_  B
)
8 inss2 3644 . . . . . . 7  |-  ( ~P A  i^i  Fin )  C_ 
Fin
9 simpr 468 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  b  e.  ( ~P A  i^i  Fin ) )
108, 9sseldi 3416 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  b  e.  Fin )
11 f1ofun 5830 . . . . . . . 8  |-  ( F : A -1-1-onto-> B  ->  Fun  F )
1211adantr 472 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  Fun  F )
13 inss1 3643 . . . . . . . . . . 11  |-  ( ~P A  i^i  Fin )  C_ 
~P A
1413sseli 3414 . . . . . . . . . 10  |-  ( b  e.  ( ~P A  i^i  Fin )  ->  b  e.  ~P A )
15 elpwi 3951 . . . . . . . . . 10  |-  ( b  e.  ~P A  -> 
b  C_  A )
1614, 15syl 17 . . . . . . . . 9  |-  ( b  e.  ( ~P A  i^i  Fin )  ->  b  C_  A )
1716adantl 473 . . . . . . . 8  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  b  C_  A
)
18 f1odm 5832 . . . . . . . . 9  |-  ( F : A -1-1-onto-> B  ->  dom  F  =  A )
1918adantr 472 . . . . . . . 8  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  dom  F  =  A )
2017, 19sseqtr4d 3455 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  b  C_  dom  F )
21 fores 5815 . . . . . . 7  |-  ( ( Fun  F  /\  b  C_ 
dom  F )  -> 
( F  |`  b
) : b -onto-> ( F " b ) )
2212, 20, 21syl2anc 673 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F  |`  b ) : b
-onto-> ( F " b
) )
23 fofi 7878 . . . . . 6  |-  ( ( b  e.  Fin  /\  ( F  |`  b ) : b -onto-> ( F
" b ) )  ->  ( F "
b )  e.  Fin )
2410, 22, 23syl2anc 673 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F "
b )  e.  Fin )
25 elpwg 3950 . . . . 5  |-  ( ( F " b )  e.  Fin  ->  (
( F " b
)  e.  ~P B  <->  ( F " b ) 
C_  B ) )
2624, 25syl 17 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( ( F
" b )  e. 
~P B  <->  ( F " b )  C_  B
) )
277, 26mpbird 240 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F "
b )  e.  ~P B )
2827, 24elind 3609 . 2  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F "
b )  e.  ( ~P B  i^i  Fin ) )
29 imassrn 5185 . . . . . 6  |-  ( `' F " a ) 
C_  ran  `' F
30 dfdm4 5032 . . . . . . 7  |-  dom  F  =  ran  `' F
3130, 18syl5eqr 2519 . . . . . 6  |-  ( F : A -1-1-onto-> B  ->  ran  `' F  =  A )
3229, 31syl5sseq 3466 . . . . 5  |-  ( F : A -1-1-onto-> B  ->  ( `' F " a )  C_  A )
3332adantr 472 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F " a )  C_  A
)
34 inss2 3644 . . . . . . 7  |-  ( ~P B  i^i  Fin )  C_ 
Fin
35 simpr 468 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  e.  ( ~P B  i^i  Fin ) )
3634, 35sseldi 3416 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  e.  Fin )
37 dff1o3 5834 . . . . . . . . 9  |-  ( F : A -1-1-onto-> B  <->  ( F : A -onto-> B  /\  Fun  `' F ) )
3837simprbi 471 . . . . . . . 8  |-  ( F : A -1-1-onto-> B  ->  Fun  `' F
)
3938adantr 472 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  Fun  `' F
)
40 inss1 3643 . . . . . . . . . . 11  |-  ( ~P B  i^i  Fin )  C_ 
~P B
4140sseli 3414 . . . . . . . . . 10  |-  ( a  e.  ( ~P B  i^i  Fin )  ->  a  e.  ~P B )
4241adantl 473 . . . . . . . . 9  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  e.  ~P B )
43 elpwi 3951 . . . . . . . . 9  |-  ( a  e.  ~P B  -> 
a  C_  B )
4442, 43syl 17 . . . . . . . 8  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  C_  B
)
45 f1ocnv 5840 . . . . . . . . . 10  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
4645adantr 472 . . . . . . . . 9  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  `' F : B
-1-1-onto-> A )
47 f1odm 5832 . . . . . . . . 9  |-  ( `' F : B -1-1-onto-> A  ->  dom  `' F  =  B
)
4846, 47syl 17 . . . . . . . 8  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  dom  `' F  =  B )
4944, 48sseqtr4d 3455 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  C_  dom  `' F )
50 fores 5815 . . . . . . 7  |-  ( ( Fun  `' F  /\  a  C_  dom  `' F
)  ->  ( `' F  |`  a ) : a -onto-> ( `' F " a ) )
5139, 49, 50syl2anc 673 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F  |`  a ) : a
-onto-> ( `' F "
a ) )
52 fofi 7878 . . . . . 6  |-  ( ( a  e.  Fin  /\  ( `' F  |`  a ) : a -onto-> ( `' F " a ) )  ->  ( `' F " a )  e. 
Fin )
5336, 51, 52syl2anc 673 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F " a )  e.  Fin )
54 elpwg 3950 . . . . 5  |-  ( ( `' F " a )  e.  Fin  ->  (
( `' F "
a )  e.  ~P A 
<->  ( `' F "
a )  C_  A
) )
5553, 54syl 17 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( ( `' F " a )  e.  ~P A  <->  ( `' F " a )  C_  A ) )
5633, 55mpbird 240 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F " a )  e.  ~P A )
5756, 53elind 3609 . 2  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F " a )  e.  ( ~P A  i^i  Fin ) )
5814, 41anim12i 576 . . 3  |-  ( ( b  e.  ( ~P A  i^i  Fin )  /\  a  e.  ( ~P B  i^i  Fin )
)  ->  ( b  e.  ~P A  /\  a  e.  ~P B ) )
5943adantl 473 . . . . . . 7  |-  ( ( b  e.  ~P A  /\  a  e.  ~P B )  ->  a  C_  B )
60 foimacnv 5845 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  a  C_  B )  ->  ( F "
( `' F "
a ) )  =  a )
613, 59, 60syl2an 485 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( F " ( `' F " a ) )  =  a )
6261eqcomd 2477 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
a  =  ( F
" ( `' F " a ) ) )
63 imaeq2 5170 . . . . . 6  |-  ( b  =  ( `' F " a )  ->  ( F " b )  =  ( F " ( `' F " a ) ) )
6463eqeq2d 2481 . . . . 5  |-  ( b  =  ( `' F " a )  ->  (
a  =  ( F
" b )  <->  a  =  ( F " ( `' F " a ) ) ) )
6562, 64syl5ibrcom 230 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( b  =  ( `' F " a )  ->  a  =  ( F " b ) ) )
66 f1of1 5827 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -1-1-> B )
6715adantr 472 . . . . . . 7  |-  ( ( b  e.  ~P A  /\  a  e.  ~P B )  ->  b  C_  A )
68 f1imacnv 5844 . . . . . . 7  |-  ( ( F : A -1-1-> B  /\  b  C_  A )  ->  ( `' F " ( F " b
) )  =  b )
6966, 67, 68syl2an 485 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( `' F "
( F " b
) )  =  b )
7069eqcomd 2477 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
b  =  ( `' F " ( F
" b ) ) )
71 imaeq2 5170 . . . . . 6  |-  ( a  =  ( F "
b )  ->  ( `' F " a )  =  ( `' F " ( F " b
) ) )
7271eqeq2d 2481 . . . . 5  |-  ( a  =  ( F "
b )  ->  (
b  =  ( `' F " a )  <-> 
b  =  ( `' F " ( F
" b ) ) ) )
7370, 72syl5ibrcom 230 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( a  =  ( F " b )  ->  b  =  ( `' F " a ) ) )
7465, 73impbid 195 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( b  =  ( `' F " a )  <-> 
a  =  ( F
" b ) ) )
7558, 74sylan2 482 . 2  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ( ~P A  i^i  Fin )  /\  a  e.  ( ~P B  i^i  Fin ) ) )  -> 
( b  =  ( `' F " a )  <-> 
a  =  ( F
" b ) ) )
761, 28, 57, 75f1o2d 6540 1  |-  ( F : A -1-1-onto-> B  ->  ( b  e.  ( ~P A  i^i  Fin )  |->  ( F "
b ) ) : ( ~P A  i^i  Fin ) -1-1-onto-> ( ~P B  i^i  Fin ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    i^i cin 3389    C_ wss 3390   ~Pcpw 3942    |-> cmpt 4454   `'ccnv 4838   dom cdm 4839   ran crn 4840    |` cres 4841   "cima 4842   Fun wfun 5583   -1-1->wf1 5586   -onto->wfo 5587   -1-1-onto->wf1o 5588   Fincfn 7587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-1o 7200  df-er 7381  df-en 7588  df-dom 7589  df-fin 7591
This theorem is referenced by:  fictb  8693  ackbijnn  13963  tsmsf1o  21237  eulerpartgbij  29278
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