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Theorem restfpw 19474
Description: The restriction of the set of finite subsets of  A is the set of finite subsets of  B. (Contributed by Mario Carneiro, 18-Sep-2015.)
Assertion
Ref Expression
restfpw  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( ( ~P A  i^i  Fin )t  B )  =  ( ~P B  i^i  Fin ) )

Proof of Theorem restfpw
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pwexg 4631 . . . . . 6  |-  ( A  e.  V  ->  ~P A  e.  _V )
21adantr 465 . . . . 5  |-  ( ( A  e.  V  /\  B  C_  A )  ->  ~P A  e.  _V )
3 inex1g 4590 . . . . 5  |-  ( ~P A  e.  _V  ->  ( ~P A  i^i  Fin )  e.  _V )
42, 3syl 16 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( ~P A  i^i  Fin )  e.  _V )
5 ssexg 4593 . . . . 5  |-  ( ( B  C_  A  /\  A  e.  V )  ->  B  e.  _V )
65ancoms 453 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A )  ->  B  e.  _V )
7 restval 14682 . . . 4  |-  ( ( ( ~P A  i^i  Fin )  e.  _V  /\  B  e.  _V )  ->  ( ( ~P A  i^i  Fin )t  B )  =  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( x  i^i  B
) ) )
84, 6, 7syl2anc 661 . . 3  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( ( ~P A  i^i  Fin )t  B )  =  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( x  i^i  B
) ) )
9 inss2 3719 . . . . . . 7  |-  ( x  i^i  B )  C_  B
109a1i 11 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
x  i^i  B )  C_  B )
11 elfpw 7822 . . . . . . . . 9  |-  ( x  e.  ( ~P A  i^i  Fin )  <->  ( x  C_  A  /\  x  e. 
Fin ) )
1211simprbi 464 . . . . . . . 8  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  Fin )
1312adantl 466 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  Fin )
14 inss1 3718 . . . . . . 7  |-  ( x  i^i  B )  C_  x
15 ssfi 7740 . . . . . . 7  |-  ( ( x  e.  Fin  /\  ( x  i^i  B ) 
C_  x )  -> 
( x  i^i  B
)  e.  Fin )
1613, 14, 15sylancl 662 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
x  i^i  B )  e.  Fin )
17 elfpw 7822 . . . . . 6  |-  ( ( x  i^i  B )  e.  ( ~P B  i^i  Fin )  <->  ( (
x  i^i  B )  C_  B  /\  ( x  i^i  B )  e. 
Fin ) )
1810, 16, 17sylanbrc 664 . . . . 5  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
x  i^i  B )  e.  ( ~P B  i^i  Fin ) )
19 eqid 2467 . . . . 5  |-  ( x  e.  ( ~P A  i^i  Fin )  |->  ( x  i^i  B ) )  =  ( x  e.  ( ~P A  i^i  Fin )  |->  ( x  i^i 
B ) )
2018, 19fmptd 6045 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( x  e.  ( ~P A  i^i  Fin )  |->  ( x  i^i 
B ) ) : ( ~P A  i^i  Fin ) --> ( ~P B  i^i  Fin ) )
21 frn 5737 . . . 4  |-  ( ( x  e.  ( ~P A  i^i  Fin )  |->  ( x  i^i  B
) ) : ( ~P A  i^i  Fin )
--> ( ~P B  i^i  Fin )  ->  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( x  i^i  B ) ) 
C_  ( ~P B  i^i  Fin ) )
2220, 21syl 16 . . 3  |-  ( ( A  e.  V  /\  B  C_  A )  ->  ran  ( x  e.  ( ~P A  i^i  Fin )  |->  ( x  i^i 
B ) )  C_  ( ~P B  i^i  Fin ) )
238, 22eqsstrd 3538 . 2  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( ( ~P A  i^i  Fin )t  B )  C_  ( ~P B  i^i  Fin )
)
24 elfpw 7822 . . . . . . . 8  |-  ( x  e.  ( ~P B  i^i  Fin )  <->  ( x  C_  B  /\  x  e. 
Fin ) )
2524simplbi 460 . . . . . . 7  |-  ( x  e.  ( ~P B  i^i  Fin )  ->  x  C_  B )
2625adantl 466 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  x  C_  B )
27 df-ss 3490 . . . . . 6  |-  ( x 
C_  B  <->  ( x  i^i  B )  =  x )
2826, 27sylib 196 . . . . 5  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  (
x  i^i  B )  =  x )
294adantr 465 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  ( ~P A  i^i  Fin )  e.  _V )
306adantr 465 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  B  e.  _V )
31 simplr 754 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  B  C_  A )
3226, 31sstrd 3514 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  x  C_  A )
3324simprbi 464 . . . . . . . 8  |-  ( x  e.  ( ~P B  i^i  Fin )  ->  x  e.  Fin )
3433adantl 466 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  x  e.  Fin )
3532, 34, 11sylanbrc 664 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  x  e.  ( ~P A  i^i  Fin ) )
36 elrestr 14684 . . . . . 6  |-  ( ( ( ~P A  i^i  Fin )  e.  _V  /\  B  e.  _V  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
x  i^i  B )  e.  ( ( ~P A  i^i  Fin )t  B ) )
3729, 30, 35, 36syl3anc 1228 . . . . 5  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  (
x  i^i  B )  e.  ( ( ~P A  i^i  Fin )t  B ) )
3828, 37eqeltrrd 2556 . . . 4  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  x  e.  ( ( ~P A  i^i  Fin )t  B ) )
3938ex 434 . . 3  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( x  e.  ( ~P B  i^i  Fin )  ->  x  e.  ( ( ~P A  i^i  Fin )t  B ) ) )
4039ssrdv 3510 . 2  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( ~P B  i^i  Fin )  C_  ( ( ~P A  i^i  Fin )t  B
) )
4123, 40eqssd 3521 1  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( ( ~P A  i^i  Fin )t  B )  =  ( ~P B  i^i  Fin ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    i^i cin 3475    C_ wss 3476   ~Pcpw 4010    |-> cmpt 4505   ran crn 5000   -->wf 5584  (class class class)co 6284   Fincfn 7516   ↾t crest 14676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-er 7311  df-en 7517  df-fin 7520  df-rest 14678
This theorem is referenced by: (None)
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