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Theorem restfpw 19975
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 4580 . . . . . 6  |-  ( A  e.  V  ->  ~P A  e.  _V )
21adantr 465 . . . . 5  |-  ( ( A  e.  V  /\  B  C_  A )  ->  ~P A  e.  _V )
3 inex1g 4539 . . . . 5  |-  ( ~P A  e.  _V  ->  ( ~P A  i^i  Fin )  e.  _V )
42, 3syl 17 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( ~P A  i^i  Fin )  e.  _V )
5 ssexg 4542 . . . . 5  |-  ( ( B  C_  A  /\  A  e.  V )  ->  B  e.  _V )
65ancoms 453 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A )  ->  B  e.  _V )
7 restval 15043 . . . 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 3662 . . . . . . 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 7858 . . . . . . . . 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 3661 . . . . . . 7  |-  ( x  i^i  B )  C_  x
15 ssfi 7777 . . . . . . 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 7858 . . . . . 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 2404 . . . . 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 6035 . . . 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 5722 . . . 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 17 . . 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 3478 . 2  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( ( ~P A  i^i  Fin )t  B )  C_  ( ~P B  i^i  Fin )
)
24 elfpw 7858 . . . . . . . 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 3430 . . . . . 6  |-  ( x 
C_  B  <->  ( x  i^i  B )  =  x )
2826, 27sylib 198 . . . . 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 756 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  B  C_  A )
3226, 31sstrd 3454 . . . . . . 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 15045 . . . . . 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 1232 . . . . 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 2493 . . . 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 3450 . 2  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( ~P B  i^i  Fin )  C_  ( ( ~P A  i^i  Fin )t  B
) )
4123, 40eqssd 3461 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 1407    e. wcel 1844   _Vcvv 3061    i^i cin 3415    C_ wss 3416   ~Pcpw 3957    |-> cmpt 4455   ran crn 4826   -->wf 5567  (class class class)co 6280   Fincfn 7556   ↾t crest 15037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-er 7350  df-en 7557  df-fin 7560  df-rest 15039
This theorem is referenced by: (None)
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