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Theorem fiss 6831
Description: Subset relationship for function  fi. (Contributed by Jeff Hankins, 7-Oct-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
Assertion
Ref Expression
fiss  |-  ( ( B  e.  V  /\  A  C_  B )  -> 
( fi `  A
)  C_  ( fi `  B ) )

Proof of Theorem fiss
Dummy variables  r  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 109 . . . 4  |-  ( ( B  e.  V  /\  A  C_  B )  ->  A  C_  B )
2 sspwb 4106 . . . . 5  |-  ( A 
C_  B  <->  ~P A  C_ 
~P B )
3 ssrin 3269 . . . . 5  |-  ( ~P A  C_  ~P B  ->  ( ~P A  i^i  Fin )  C_  ( ~P B  i^i  Fin ) )
42, 3sylbi 120 . . . 4  |-  ( A 
C_  B  ->  ( ~P A  i^i  Fin )  C_  ( ~P B  i^i  Fin ) )
5 ssrexv 3130 . . . 4  |-  ( ( ~P A  i^i  Fin )  C_  ( ~P B  i^i  Fin )  ->  ( E. x  e.  ( ~P A  i^i  Fin )
r  =  |^| x  ->  E. x  e.  ( ~P B  i^i  Fin ) r  =  |^| x ) )
61, 4, 53syl 17 . . 3  |-  ( ( B  e.  V  /\  A  C_  B )  -> 
( E. x  e.  ( ~P A  i^i  Fin ) r  =  |^| x  ->  E. x  e.  ( ~P B  i^i  Fin ) r  =  |^| x ) )
7 vex 2661 . . . 4  |-  r  e. 
_V
8 simpl 108 . . . . 5  |-  ( ( B  e.  V  /\  A  C_  B )  ->  B  e.  V )
98, 1ssexd 4036 . . . 4  |-  ( ( B  e.  V  /\  A  C_  B )  ->  A  e.  _V )
10 elfi 6825 . . . 4  |-  ( ( r  e.  _V  /\  A  e.  _V )  ->  ( r  e.  ( fi `  A )  <->  E. x  e.  ( ~P A  i^i  Fin )
r  =  |^| x
) )
117, 9, 10sylancr 408 . . 3  |-  ( ( B  e.  V  /\  A  C_  B )  -> 
( r  e.  ( fi `  A )  <->  E. x  e.  ( ~P A  i^i  Fin )
r  =  |^| x
) )
12 elfi 6825 . . . . 5  |-  ( ( r  e.  _V  /\  B  e.  V )  ->  ( r  e.  ( fi `  B )  <->  E. x  e.  ( ~P B  i^i  Fin )
r  =  |^| x
) )
137, 12mpan 418 . . . 4  |-  ( B  e.  V  ->  (
r  e.  ( fi
`  B )  <->  E. x  e.  ( ~P B  i^i  Fin ) r  =  |^| x ) )
1413adantr 272 . . 3  |-  ( ( B  e.  V  /\  A  C_  B )  -> 
( r  e.  ( fi `  B )  <->  E. x  e.  ( ~P B  i^i  Fin )
r  =  |^| x
) )
156, 11, 143imtr4d 202 . 2  |-  ( ( B  e.  V  /\  A  C_  B )  -> 
( r  e.  ( fi `  A )  ->  r  e.  ( fi `  B ) ) )
1615ssrdv 3071 1  |-  ( ( B  e.  V  /\  A  C_  B )  -> 
( fi `  A
)  C_  ( fi `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   E.wrex 2392   _Vcvv 2658    i^i cin 3038    C_ wss 3039   ~Pcpw 3478   |^|cint 3739   ` cfv 5091   Fincfn 6600   ficfi 6822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-er 6395  df-en 6601  df-fin 6603  df-fi 6823
This theorem is referenced by: (None)
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