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Theorem f1opwfi 7636
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 2443 . 2  |-  ( b  e.  ( ~P A  i^i  Fin )  |->  ( F
" b ) )  =  ( b  e.  ( ~P A  i^i  Fin )  |->  ( F "
b ) )
2 imassrn 5201 . . . . . 6  |-  ( F
" b )  C_  ran  F
3 f1ofo 5669 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -onto-> B )
4 forn 5644 . . . . . . 7  |-  ( F : A -onto-> B  ->  ran  F  =  B )
53, 4syl 16 . . . . . 6  |-  ( F : A -1-1-onto-> B  ->  ran  F  =  B )
62, 5syl5sseq 3425 . . . . 5  |-  ( F : A -1-1-onto-> B  ->  ( F " b )  C_  B
)
76adantr 465 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F "
b )  C_  B
)
8 inss2 3592 . . . . . . 7  |-  ( ~P A  i^i  Fin )  C_ 
Fin
9 simpr 461 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  b  e.  ( ~P A  i^i  Fin ) )
108, 9sseldi 3375 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  b  e.  Fin )
11 f1ofun 5664 . . . . . . . 8  |-  ( F : A -1-1-onto-> B  ->  Fun  F )
1211adantr 465 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  Fun  F )
13 inss1 3591 . . . . . . . . . . 11  |-  ( ~P A  i^i  Fin )  C_ 
~P A
1413sseli 3373 . . . . . . . . . 10  |-  ( b  e.  ( ~P A  i^i  Fin )  ->  b  e.  ~P A )
15 elpwi 3890 . . . . . . . . . 10  |-  ( b  e.  ~P A  -> 
b  C_  A )
1614, 15syl 16 . . . . . . . . 9  |-  ( b  e.  ( ~P A  i^i  Fin )  ->  b  C_  A )
1716adantl 466 . . . . . . . 8  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  b  C_  A
)
18 f1odm 5666 . . . . . . . . 9  |-  ( F : A -1-1-onto-> B  ->  dom  F  =  A )
1918adantr 465 . . . . . . . 8  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  dom  F  =  A )
2017, 19sseqtr4d 3414 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  b  C_  dom  F )
21 fores 5650 . . . . . . 7  |-  ( ( Fun  F  /\  b  C_ 
dom  F )  -> 
( F  |`  b
) : b -onto-> ( F " b ) )
2212, 20, 21syl2anc 661 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F  |`  b ) : b
-onto-> ( F " b
) )
23 fofi 7618 . . . . . 6  |-  ( ( b  e.  Fin  /\  ( F  |`  b ) : b -onto-> ( F
" b ) )  ->  ( F "
b )  e.  Fin )
2410, 22, 23syl2anc 661 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F "
b )  e.  Fin )
25 elpwg 3889 . . . . 5  |-  ( ( F " b )  e.  Fin  ->  (
( F " b
)  e.  ~P B  <->  ( F " b ) 
C_  B ) )
2624, 25syl 16 . . . 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 232 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F "
b )  e.  ~P B )
2827, 24elind 3561 . 2  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F "
b )  e.  ( ~P B  i^i  Fin ) )
29 imassrn 5201 . . . . . 6  |-  ( `' F " a ) 
C_  ran  `' F
30 dfdm4 5053 . . . . . . 7  |-  dom  F  =  ran  `' F
3130, 18syl5eqr 2489 . . . . . 6  |-  ( F : A -1-1-onto-> B  ->  ran  `' F  =  A )
3229, 31syl5sseq 3425 . . . . 5  |-  ( F : A -1-1-onto-> B  ->  ( `' F " a )  C_  A )
3332adantr 465 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F " a )  C_  A
)
34 inss2 3592 . . . . . . 7  |-  ( ~P B  i^i  Fin )  C_ 
Fin
35 simpr 461 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  e.  ( ~P B  i^i  Fin ) )
3634, 35sseldi 3375 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  e.  Fin )
37 dff1o3 5668 . . . . . . . . 9  |-  ( F : A -1-1-onto-> B  <->  ( F : A -onto-> B  /\  Fun  `' F ) )
3837simprbi 464 . . . . . . . 8  |-  ( F : A -1-1-onto-> B  ->  Fun  `' F
)
3938adantr 465 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  Fun  `' F
)
40 inss1 3591 . . . . . . . . . . 11  |-  ( ~P B  i^i  Fin )  C_ 
~P B
4140sseli 3373 . . . . . . . . . 10  |-  ( a  e.  ( ~P B  i^i  Fin )  ->  a  e.  ~P B )
4241adantl 466 . . . . . . . . 9  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  e.  ~P B )
43 elpwi 3890 . . . . . . . . 9  |-  ( a  e.  ~P B  -> 
a  C_  B )
4442, 43syl 16 . . . . . . . 8  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  C_  B
)
45 f1ocnv 5674 . . . . . . . . . 10  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
4645adantr 465 . . . . . . . . 9  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  `' F : B
-1-1-onto-> A )
47 f1odm 5666 . . . . . . . . 9  |-  ( `' F : B -1-1-onto-> A  ->  dom  `' F  =  B
)
4846, 47syl 16 . . . . . . . 8  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  dom  `' F  =  B )
4944, 48sseqtr4d 3414 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  C_  dom  `' F )
50 fores 5650 . . . . . . 7  |-  ( ( Fun  `' F  /\  a  C_  dom  `' F
)  ->  ( `' F  |`  a ) : a -onto-> ( `' F " a ) )
5139, 49, 50syl2anc 661 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F  |`  a ) : a
-onto-> ( `' F "
a ) )
52 fofi 7618 . . . . . 6  |-  ( ( a  e.  Fin  /\  ( `' F  |`  a ) : a -onto-> ( `' F " a ) )  ->  ( `' F " a )  e. 
Fin )
5336, 51, 52syl2anc 661 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F " a )  e.  Fin )
54 elpwg 3889 . . . . 5  |-  ( ( `' F " a )  e.  Fin  ->  (
( `' F "
a )  e.  ~P A 
<->  ( `' F "
a )  C_  A
) )
5553, 54syl 16 . . . 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 232 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F " a )  e.  ~P A )
5756, 53elind 3561 . 2  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F " a )  e.  ( ~P A  i^i  Fin ) )
5814, 41anim12i 566 . . 3  |-  ( ( b  e.  ( ~P A  i^i  Fin )  /\  a  e.  ( ~P B  i^i  Fin )
)  ->  ( b  e.  ~P A  /\  a  e.  ~P B ) )
5943adantl 466 . . . . . . 7  |-  ( ( b  e.  ~P A  /\  a  e.  ~P B )  ->  a  C_  B )
60 foimacnv 5679 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  a  C_  B )  ->  ( F "
( `' F "
a ) )  =  a )
613, 59, 60syl2an 477 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( F " ( `' F " a ) )  =  a )
6261eqcomd 2448 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
a  =  ( F
" ( `' F " a ) ) )
63 imaeq2 5186 . . . . . 6  |-  ( b  =  ( `' F " a )  ->  ( F " b )  =  ( F " ( `' F " a ) ) )
6463eqeq2d 2454 . . . . 5  |-  ( b  =  ( `' F " a )  ->  (
a  =  ( F
" b )  <->  a  =  ( F " ( `' F " a ) ) ) )
6562, 64syl5ibrcom 222 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( b  =  ( `' F " a )  ->  a  =  ( F " b ) ) )
66 f1of1 5661 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -1-1-> B )
6715adantr 465 . . . . . . 7  |-  ( ( b  e.  ~P A  /\  a  e.  ~P B )  ->  b  C_  A )
68 f1imacnv 5678 . . . . . . 7  |-  ( ( F : A -1-1-> B  /\  b  C_  A )  ->  ( `' F " ( F " b
) )  =  b )
6966, 67, 68syl2an 477 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( `' F "
( F " b
) )  =  b )
7069eqcomd 2448 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
b  =  ( `' F " ( F
" b ) ) )
71 imaeq2 5186 . . . . . 6  |-  ( a  =  ( F "
b )  ->  ( `' F " a )  =  ( `' F " ( F " b
) ) )
7271eqeq2d 2454 . . . . 5  |-  ( a  =  ( F "
b )  ->  (
b  =  ( `' F " a )  <-> 
b  =  ( `' F " ( F
" b ) ) ) )
7370, 72syl5ibrcom 222 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( a  =  ( F " b )  ->  b  =  ( `' F " a ) ) )
7465, 73impbid 191 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( b  =  ( `' F " a )  <-> 
a  =  ( F
" b ) ) )
7558, 74sylan2 474 . 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 6333 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 184    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3348    C_ wss 3349   ~Pcpw 3881    e. cmpt 4371   `'ccnv 4860   dom cdm 4861   ran crn 4862    |` cres 4863   "cima 4864   Fun wfun 5433   -1-1->wf1 5436   -onto->wfo 5437   -1-1-onto->wf1o 5438   Fincfn 7331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-om 6498  df-1o 6941  df-er 7122  df-en 7332  df-dom 7333  df-fin 7335
This theorem is referenced by:  fictb  8435  ackbijnn  13312  tsmsf1o  19741  eulerpartgbij  26777
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