MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1opwfi Unicode version

Theorem f1opwfi 7175
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 2296 . 2  |-  ( b  e.  ( ~P A  i^i  Fin )  |->  ( F
" b ) )  =  ( b  e.  ( ~P A  i^i  Fin )  |->  ( F "
b ) )
2 imassrn 5041 . . . . . 6  |-  ( F
" b )  C_  ran  F
3 f1ofo 5495 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -onto-> B )
4 forn 5470 . . . . . . 7  |-  ( F : A -onto-> B  ->  ran  F  =  B )
53, 4syl 15 . . . . . 6  |-  ( F : A -1-1-onto-> B  ->  ran  F  =  B )
62, 5syl5sseq 3239 . . . . 5  |-  ( F : A -1-1-onto-> B  ->  ( F " b )  C_  B
)
76adantr 451 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F "
b )  C_  B
)
8 inss2 3403 . . . . . . 7  |-  ( ~P A  i^i  Fin )  C_ 
Fin
9 simpr 447 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  b  e.  ( ~P A  i^i  Fin ) )
108, 9sseldi 3191 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  b  e.  Fin )
11 f1ofun 5490 . . . . . . . 8  |-  ( F : A -1-1-onto-> B  ->  Fun  F )
1211adantr 451 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  Fun  F )
13 inss1 3402 . . . . . . . . . . 11  |-  ( ~P A  i^i  Fin )  C_ 
~P A
1413sseli 3189 . . . . . . . . . 10  |-  ( b  e.  ( ~P A  i^i  Fin )  ->  b  e.  ~P A )
15 elpwi 3646 . . . . . . . . . 10  |-  ( b  e.  ~P A  -> 
b  C_  A )
1614, 15syl 15 . . . . . . . . 9  |-  ( b  e.  ( ~P A  i^i  Fin )  ->  b  C_  A )
1716adantl 452 . . . . . . . 8  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  b  C_  A
)
18 f1odm 5492 . . . . . . . . 9  |-  ( F : A -1-1-onto-> B  ->  dom  F  =  A )
1918adantr 451 . . . . . . . 8  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  dom  F  =  A )
2017, 19sseqtr4d 3228 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  b  C_  dom  F )
21 fores 5476 . . . . . . 7  |-  ( ( Fun  F  /\  b  C_ 
dom  F )  -> 
( F  |`  b
) : b -onto-> ( F " b ) )
2212, 20, 21syl2anc 642 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F  |`  b ) : b
-onto-> ( F " b
) )
23 fofi 7158 . . . . . 6  |-  ( ( b  e.  Fin  /\  ( F  |`  b ) : b -onto-> ( F
" b ) )  ->  ( F "
b )  e.  Fin )
2410, 22, 23syl2anc 642 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F "
b )  e.  Fin )
25 elpwg 3645 . . . . 5  |-  ( ( F " b )  e.  Fin  ->  (
( F " b
)  e.  ~P B  <->  ( F " b ) 
C_  B ) )
2624, 25syl 15 . . . 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 223 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F "
b )  e.  ~P B )
28 elin 3371 . . 3  |-  ( ( F " b )  e.  ( ~P B  i^i  Fin )  <->  ( ( F " b )  e. 
~P B  /\  ( F " b )  e. 
Fin ) )
2927, 24, 28sylanbrc 645 . 2  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F "
b )  e.  ( ~P B  i^i  Fin ) )
30 imassrn 5041 . . . . . 6  |-  ( `' F " a ) 
C_  ran  `' F
31 dfdm4 4888 . . . . . . 7  |-  dom  F  =  ran  `' F
3231, 18syl5eqr 2342 . . . . . 6  |-  ( F : A -1-1-onto-> B  ->  ran  `' F  =  A )
3330, 32syl5sseq 3239 . . . . 5  |-  ( F : A -1-1-onto-> B  ->  ( `' F " a )  C_  A )
3433adantr 451 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F " a )  C_  A
)
35 inss2 3403 . . . . . . 7  |-  ( ~P B  i^i  Fin )  C_ 
Fin
36 simpr 447 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  e.  ( ~P B  i^i  Fin ) )
3735, 36sseldi 3191 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  e.  Fin )
38 dff1o3 5494 . . . . . . . . 9  |-  ( F : A -1-1-onto-> B  <->  ( F : A -onto-> B  /\  Fun  `' F ) )
3938simprbi 450 . . . . . . . 8  |-  ( F : A -1-1-onto-> B  ->  Fun  `' F
)
4039adantr 451 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  Fun  `' F
)
41 inss1 3402 . . . . . . . . . . 11  |-  ( ~P B  i^i  Fin )  C_ 
~P B
4241sseli 3189 . . . . . . . . . 10  |-  ( a  e.  ( ~P B  i^i  Fin )  ->  a  e.  ~P B )
4342adantl 452 . . . . . . . . 9  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  e.  ~P B )
44 elpwi 3646 . . . . . . . . 9  |-  ( a  e.  ~P B  -> 
a  C_  B )
4543, 44syl 15 . . . . . . . 8  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  C_  B
)
46 f1ocnv 5501 . . . . . . . . . 10  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
4746adantr 451 . . . . . . . . 9  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  `' F : B
-1-1-onto-> A )
48 f1odm 5492 . . . . . . . . 9  |-  ( `' F : B -1-1-onto-> A  ->  dom  `' F  =  B
)
4947, 48syl 15 . . . . . . . 8  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  dom  `' F  =  B )
5045, 49sseqtr4d 3228 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  C_  dom  `' F )
51 fores 5476 . . . . . . 7  |-  ( ( Fun  `' F  /\  a  C_  dom  `' F
)  ->  ( `' F  |`  a ) : a -onto-> ( `' F " a ) )
5240, 50, 51syl2anc 642 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F  |`  a ) : a
-onto-> ( `' F "
a ) )
53 fofi 7158 . . . . . 6  |-  ( ( a  e.  Fin  /\  ( `' F  |`  a ) : a -onto-> ( `' F " a ) )  ->  ( `' F " a )  e. 
Fin )
5437, 52, 53syl2anc 642 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F " a )  e.  Fin )
55 elpwg 3645 . . . . 5  |-  ( ( `' F " a )  e.  Fin  ->  (
( `' F "
a )  e.  ~P A 
<->  ( `' F "
a )  C_  A
) )
5654, 55syl 15 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( ( `' F " a )  e.  ~P A  <->  ( `' F " a )  C_  A ) )
5734, 56mpbird 223 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F " a )  e.  ~P A )
58 elin 3371 . . 3  |-  ( ( `' F " a )  e.  ( ~P A  i^i  Fin )  <->  ( ( `' F " a )  e.  ~P A  /\  ( `' F " a )  e.  Fin ) )
5957, 54, 58sylanbrc 645 . 2  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F " a )  e.  ( ~P A  i^i  Fin ) )
6014, 42anim12i 549 . . 3  |-  ( ( b  e.  ( ~P A  i^i  Fin )  /\  a  e.  ( ~P B  i^i  Fin )
)  ->  ( b  e.  ~P A  /\  a  e.  ~P B ) )
6144adantl 452 . . . . . . 7  |-  ( ( b  e.  ~P A  /\  a  e.  ~P B )  ->  a  C_  B )
62 foimacnv 5506 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  a  C_  B )  ->  ( F "
( `' F "
a ) )  =  a )
633, 61, 62syl2an 463 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( F " ( `' F " a ) )  =  a )
6463eqcomd 2301 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
a  =  ( F
" ( `' F " a ) ) )
65 imaeq2 5024 . . . . . 6  |-  ( b  =  ( `' F " a )  ->  ( F " b )  =  ( F " ( `' F " a ) ) )
6665eqeq2d 2307 . . . . 5  |-  ( b  =  ( `' F " a )  ->  (
a  =  ( F
" b )  <->  a  =  ( F " ( `' F " a ) ) ) )
6764, 66syl5ibrcom 213 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( b  =  ( `' F " a )  ->  a  =  ( F " b ) ) )
68 f1of1 5487 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -1-1-> B )
6915adantr 451 . . . . . . 7  |-  ( ( b  e.  ~P A  /\  a  e.  ~P B )  ->  b  C_  A )
70 f1imacnv 5505 . . . . . . 7  |-  ( ( F : A -1-1-> B  /\  b  C_  A )  ->  ( `' F " ( F " b
) )  =  b )
7168, 69, 70syl2an 463 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( `' F "
( F " b
) )  =  b )
7271eqcomd 2301 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
b  =  ( `' F " ( F
" b ) ) )
73 imaeq2 5024 . . . . . 6  |-  ( a  =  ( F "
b )  ->  ( `' F " a )  =  ( `' F " ( F " b
) ) )
7473eqeq2d 2307 . . . . 5  |-  ( a  =  ( F "
b )  ->  (
b  =  ( `' F " a )  <-> 
b  =  ( `' F " ( F
" b ) ) ) )
7572, 74syl5ibrcom 213 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( a  =  ( F " b )  ->  b  =  ( `' F " a ) ) )
7667, 75impbid 183 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( b  =  ( `' F " a )  <-> 
a  =  ( F
" b ) ) )
7760, 76sylan2 460 . 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 ) ) )
781, 29, 59, 77f1o2d 6085 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   ~Pcpw 3638    e. cmpt 4093   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265   -1-1->wf1 5268   -onto->wfo 5269   -1-1-onto->wf1o 5270   Fincfn 6879
This theorem is referenced by:  fictb  7887  ackbijnn  12302  tsmsf1o  17843
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-fin 6883
  Copyright terms: Public domain W3C validator