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Theorem f1opwfi 7412
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 2438 . 2  |-  ( b  e.  ( ~P A  i^i  Fin )  |->  ( F
" b ) )  =  ( b  e.  ( ~P A  i^i  Fin )  |->  ( F "
b ) )
2 imassrn 5218 . . . . . 6  |-  ( F
" b )  C_  ran  F
3 f1ofo 5683 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -onto-> B )
4 forn 5658 . . . . . . 7  |-  ( F : A -onto-> B  ->  ran  F  =  B )
53, 4syl 16 . . . . . 6  |-  ( F : A -1-1-onto-> B  ->  ran  F  =  B )
62, 5syl5sseq 3398 . . . . 5  |-  ( F : A -1-1-onto-> B  ->  ( F " b )  C_  B
)
76adantr 453 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F "
b )  C_  B
)
8 inss2 3564 . . . . . . 7  |-  ( ~P A  i^i  Fin )  C_ 
Fin
9 simpr 449 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  b  e.  ( ~P A  i^i  Fin ) )
108, 9sseldi 3348 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  b  e.  Fin )
11 f1ofun 5678 . . . . . . . 8  |-  ( F : A -1-1-onto-> B  ->  Fun  F )
1211adantr 453 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  Fun  F )
13 inss1 3563 . . . . . . . . . . 11  |-  ( ~P A  i^i  Fin )  C_ 
~P A
1413sseli 3346 . . . . . . . . . 10  |-  ( b  e.  ( ~P A  i^i  Fin )  ->  b  e.  ~P A )
15 elpwi 3809 . . . . . . . . . 10  |-  ( b  e.  ~P A  -> 
b  C_  A )
1614, 15syl 16 . . . . . . . . 9  |-  ( b  e.  ( ~P A  i^i  Fin )  ->  b  C_  A )
1716adantl 454 . . . . . . . 8  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  b  C_  A
)
18 f1odm 5680 . . . . . . . . 9  |-  ( F : A -1-1-onto-> B  ->  dom  F  =  A )
1918adantr 453 . . . . . . . 8  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  dom  F  =  A )
2017, 19sseqtr4d 3387 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  b  C_  dom  F )
21 fores 5664 . . . . . . 7  |-  ( ( Fun  F  /\  b  C_ 
dom  F )  -> 
( F  |`  b
) : b -onto-> ( F " b ) )
2212, 20, 21syl2anc 644 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F  |`  b ) : b
-onto-> ( F " b
) )
23 fofi 7394 . . . . . 6  |-  ( ( b  e.  Fin  /\  ( F  |`  b ) : b -onto-> ( F
" b ) )  ->  ( F "
b )  e.  Fin )
2410, 22, 23syl2anc 644 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F "
b )  e.  Fin )
25 elpwg 3808 . . . . 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 225 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F "
b )  e.  ~P B )
28 elin 3532 . . 3  |-  ( ( F " b )  e.  ( ~P B  i^i  Fin )  <->  ( ( F " b )  e. 
~P B  /\  ( F " b )  e. 
Fin ) )
2927, 24, 28sylanbrc 647 . 2  |-  ( ( F : A -1-1-onto-> B  /\  b  e.  ( ~P A  i^i  Fin ) )  ->  ( F "
b )  e.  ( ~P B  i^i  Fin ) )
30 imassrn 5218 . . . . . 6  |-  ( `' F " a ) 
C_  ran  `' F
31 dfdm4 5065 . . . . . . 7  |-  dom  F  =  ran  `' F
3231, 18syl5eqr 2484 . . . . . 6  |-  ( F : A -1-1-onto-> B  ->  ran  `' F  =  A )
3330, 32syl5sseq 3398 . . . . 5  |-  ( F : A -1-1-onto-> B  ->  ( `' F " a )  C_  A )
3433adantr 453 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F " a )  C_  A
)
35 inss2 3564 . . . . . . 7  |-  ( ~P B  i^i  Fin )  C_ 
Fin
36 simpr 449 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  e.  ( ~P B  i^i  Fin ) )
3735, 36sseldi 3348 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  e.  Fin )
38 dff1o3 5682 . . . . . . . . 9  |-  ( F : A -1-1-onto-> B  <->  ( F : A -onto-> B  /\  Fun  `' F ) )
3938simprbi 452 . . . . . . . 8  |-  ( F : A -1-1-onto-> B  ->  Fun  `' F
)
4039adantr 453 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  Fun  `' F
)
41 inss1 3563 . . . . . . . . . . 11  |-  ( ~P B  i^i  Fin )  C_ 
~P B
4241sseli 3346 . . . . . . . . . 10  |-  ( a  e.  ( ~P B  i^i  Fin )  ->  a  e.  ~P B )
4342adantl 454 . . . . . . . . 9  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  e.  ~P B )
44 elpwi 3809 . . . . . . . . 9  |-  ( a  e.  ~P B  -> 
a  C_  B )
4543, 44syl 16 . . . . . . . 8  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  C_  B
)
46 f1ocnv 5689 . . . . . . . . . 10  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
4746adantr 453 . . . . . . . . 9  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  `' F : B
-1-1-onto-> A )
48 f1odm 5680 . . . . . . . . 9  |-  ( `' F : B -1-1-onto-> A  ->  dom  `' F  =  B
)
4947, 48syl 16 . . . . . . . 8  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  dom  `' F  =  B )
5045, 49sseqtr4d 3387 . . . . . . 7  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  a  C_  dom  `' F )
51 fores 5664 . . . . . . 7  |-  ( ( Fun  `' F  /\  a  C_  dom  `' F
)  ->  ( `' F  |`  a ) : a -onto-> ( `' F " a ) )
5240, 50, 51syl2anc 644 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F  |`  a ) : a
-onto-> ( `' F "
a ) )
53 fofi 7394 . . . . . 6  |-  ( ( a  e.  Fin  /\  ( `' F  |`  a ) : a -onto-> ( `' F " a ) )  ->  ( `' F " a )  e. 
Fin )
5437, 52, 53syl2anc 644 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F " a )  e.  Fin )
55 elpwg 3808 . . . . 5  |-  ( ( `' F " a )  e.  Fin  ->  (
( `' F "
a )  e.  ~P A 
<->  ( `' F "
a )  C_  A
) )
5654, 55syl 16 . . . 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 225 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F " a )  e.  ~P A )
58 elin 3532 . . 3  |-  ( ( `' F " a )  e.  ( ~P A  i^i  Fin )  <->  ( ( `' F " a )  e.  ~P A  /\  ( `' F " a )  e.  Fin ) )
5957, 54, 58sylanbrc 647 . 2  |-  ( ( F : A -1-1-onto-> B  /\  a  e.  ( ~P B  i^i  Fin ) )  ->  ( `' F " a )  e.  ( ~P A  i^i  Fin ) )
6014, 42anim12i 551 . . 3  |-  ( ( b  e.  ( ~P A  i^i  Fin )  /\  a  e.  ( ~P B  i^i  Fin )
)  ->  ( b  e.  ~P A  /\  a  e.  ~P B ) )
6144adantl 454 . . . . . . 7  |-  ( ( b  e.  ~P A  /\  a  e.  ~P B )  ->  a  C_  B )
62 foimacnv 5694 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  a  C_  B )  ->  ( F "
( `' F "
a ) )  =  a )
633, 61, 62syl2an 465 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( F " ( `' F " a ) )  =  a )
6463eqcomd 2443 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
a  =  ( F
" ( `' F " a ) ) )
65 imaeq2 5201 . . . . . 6  |-  ( b  =  ( `' F " a )  ->  ( F " b )  =  ( F " ( `' F " a ) ) )
6665eqeq2d 2449 . . . . 5  |-  ( b  =  ( `' F " a )  ->  (
a  =  ( F
" b )  <->  a  =  ( F " ( `' F " a ) ) ) )
6764, 66syl5ibrcom 215 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( b  =  ( `' F " a )  ->  a  =  ( F " b ) ) )
68 f1of1 5675 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -1-1-> B )
6915adantr 453 . . . . . . 7  |-  ( ( b  e.  ~P A  /\  a  e.  ~P B )  ->  b  C_  A )
70 f1imacnv 5693 . . . . . . 7  |-  ( ( F : A -1-1-> B  /\  b  C_  A )  ->  ( `' F " ( F " b
) )  =  b )
7168, 69, 70syl2an 465 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( `' F "
( F " b
) )  =  b )
7271eqcomd 2443 . . . . 5  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
b  =  ( `' F " ( F
" b ) ) )
73 imaeq2 5201 . . . . . 6  |-  ( a  =  ( F "
b )  ->  ( `' F " a )  =  ( `' F " ( F " b
) ) )
7473eqeq2d 2449 . . . . 5  |-  ( a  =  ( F "
b )  ->  (
b  =  ( `' F " a )  <-> 
b  =  ( `' F " ( F
" b ) ) ) )
7572, 74syl5ibrcom 215 . . . 4  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( a  =  ( F " b )  ->  b  =  ( `' F " a ) ) )
7667, 75impbid 185 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  ( b  e.  ~P A  /\  a  e.  ~P B ) )  -> 
( b  =  ( `' F " a )  <-> 
a  =  ( F
" b ) ) )
7760, 76sylan2 462 . 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 6298 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    i^i cin 3321    C_ wss 3322   ~Pcpw 3801    e. cmpt 4268   `'ccnv 4879   dom cdm 4880   ran crn 4881    |` cres 4882   "cima 4883   Fun wfun 5450   -1-1->wf1 5453   -onto->wfo 5454   -1-1-onto->wf1o 5455   Fincfn 7111
This theorem is referenced by:  fictb  8127  ackbijnn  12609  tsmsf1o  18176
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-1o 6726  df-er 6907  df-en 7112  df-dom 7113  df-fin 7115
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