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Theorem restfpw 20272
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 4585 . . . . . 6  |-  ( A  e.  V  ->  ~P A  e.  _V )
21adantr 472 . . . . 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 460 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A )  ->  B  e.  _V )
7 restval 15403 . . . 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 673 . . 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 3644 . . . . . . 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 7894 . . . . . . . . 9  |-  ( x  e.  ( ~P A  i^i  Fin )  <->  ( x  C_  A  /\  x  e. 
Fin ) )
1211simprbi 471 . . . . . . . 8  |-  ( x  e.  ( ~P A  i^i  Fin )  ->  x  e.  Fin )
1312adantl 473 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  x  e.  Fin )
14 inss1 3643 . . . . . . 7  |-  ( x  i^i  B )  C_  x
15 ssfi 7810 . . . . . . 7  |-  ( ( x  e.  Fin  /\  ( x  i^i  B ) 
C_  x )  -> 
( x  i^i  B
)  e.  Fin )
1613, 14, 15sylancl 675 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P A  i^i  Fin ) )  ->  (
x  i^i  B )  e.  Fin )
17 elfpw 7894 . . . . . 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 677 . . . . 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 2471 . . . . 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 6061 . . . 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 5747 . . . 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 3452 . 2  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( ( ~P A  i^i  Fin )t  B )  C_  ( ~P B  i^i  Fin )
)
24 elfpw 7894 . . . . . . . 8  |-  ( x  e.  ( ~P B  i^i  Fin )  <->  ( x  C_  B  /\  x  e. 
Fin ) )
2524simplbi 467 . . . . . . 7  |-  ( x  e.  ( ~P B  i^i  Fin )  ->  x  C_  B )
2625adantl 473 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  x  C_  B )
27 df-ss 3404 . . . . . 6  |-  ( x 
C_  B  <->  ( x  i^i  B )  =  x )
2826, 27sylib 201 . . . . 5  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  (
x  i^i  B )  =  x )
294adantr 472 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  ( ~P A  i^i  Fin )  e.  _V )
306adantr 472 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  B  e.  _V )
31 simplr 770 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  B  C_  A )
3226, 31sstrd 3428 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  x  C_  A )
3324simprbi 471 . . . . . . . 8  |-  ( x  e.  ( ~P B  i^i  Fin )  ->  x  e.  Fin )
3433adantl 473 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  x  e.  Fin )
3532, 34, 11sylanbrc 677 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  x  e.  ( ~P A  i^i  Fin ) )
36 elrestr 15405 . . . . . 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 1292 . . . . 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 2550 . . . 4  |-  ( ( ( A  e.  V  /\  B  C_  A )  /\  x  e.  ( ~P B  i^i  Fin ) )  ->  x  e.  ( ( ~P A  i^i  Fin )t  B ) )
3938ex 441 . . 3  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( x  e.  ( ~P B  i^i  Fin )  ->  x  e.  ( ( ~P A  i^i  Fin )t  B ) ) )
4039ssrdv 3424 . 2  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( ~P B  i^i  Fin )  C_  ( ( ~P A  i^i  Fin )t  B
) )
4123, 40eqssd 3435 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 376    = wceq 1452    e. wcel 1904   _Vcvv 3031    i^i cin 3389    C_ wss 3390   ~Pcpw 3942    |-> cmpt 4454   ran crn 4840   -->wf 5585  (class class class)co 6308   Fincfn 7587   ↾t crest 15397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-er 7381  df-en 7588  df-fin 7591  df-rest 15399
This theorem is referenced by: (None)
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