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Theorem fvmptss 5886
Description: If all the values of the mapping are subsets of a class  C, then so is any evaluation of the mapping, even if  D is not in the base set  A. (Contributed by Mario Carneiro, 13-Feb-2015.)
Hypothesis
Ref Expression
fvmpt2.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
fvmptss  |-  ( A. x  e.  A  B  C_  C  ->  ( F `  D )  C_  C
)
Distinct variable groups:    x, A    x, C
Allowed substitution hints:    B( x)    D( x)    F( x)

Proof of Theorem fvmptss
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fvmpt2.1 . . . . 5  |-  F  =  ( x  e.  A  |->  B )
21dmmptss 5437 . . . 4  |-  dom  F  C_  A
32sseli 3455 . . 3  |-  ( D  e.  dom  F  ->  D  e.  A )
4 fveq2 5794 . . . . . . 7  |-  ( y  =  D  ->  ( F `  y )  =  ( F `  D ) )
54sseq1d 3486 . . . . . 6  |-  ( y  =  D  ->  (
( F `  y
)  C_  C  <->  ( F `  D )  C_  C
) )
65imbi2d 316 . . . . 5  |-  ( y  =  D  ->  (
( A. x  e.  A  B  C_  C  ->  ( F `  y
)  C_  C )  <->  ( A. x  e.  A  B  C_  C  ->  ( F `  D )  C_  C ) ) )
7 nfcv 2614 . . . . . 6  |-  F/_ x
y
8 nfra1 2807 . . . . . . 7  |-  F/ x A. x  e.  A  B  C_  C
9 nfmpt1 4484 . . . . . . . . . 10  |-  F/_ x
( x  e.  A  |->  B )
101, 9nfcxfr 2612 . . . . . . . . 9  |-  F/_ x F
1110, 7nffv 5801 . . . . . . . 8  |-  F/_ x
( F `  y
)
12 nfcv 2614 . . . . . . . 8  |-  F/_ x C
1311, 12nfss 3452 . . . . . . 7  |-  F/ x
( F `  y
)  C_  C
148, 13nfim 1857 . . . . . 6  |-  F/ x
( A. x  e.  A  B  C_  C  ->  ( F `  y
)  C_  C )
15 fveq2 5794 . . . . . . . 8  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
1615sseq1d 3486 . . . . . . 7  |-  ( x  =  y  ->  (
( F `  x
)  C_  C  <->  ( F `  y )  C_  C
) )
1716imbi2d 316 . . . . . 6  |-  ( x  =  y  ->  (
( A. x  e.  A  B  C_  C  ->  ( F `  x
)  C_  C )  <->  ( A. x  e.  A  B  C_  C  ->  ( F `  y )  C_  C ) ) )
181dmmpt 5436 . . . . . . . . . . 11  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
1918rabeq2i 3069 . . . . . . . . . 10  |-  ( x  e.  dom  F  <->  ( x  e.  A  /\  B  e. 
_V ) )
201fvmpt2 5885 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  B  e.  _V )  ->  ( F `  x
)  =  B )
21 eqimss 3511 . . . . . . . . . . 11  |-  ( ( F `  x )  =  B  ->  ( F `  x )  C_  B )
2220, 21syl 16 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  B  e.  _V )  ->  ( F `  x
)  C_  B )
2319, 22sylbi 195 . . . . . . . . 9  |-  ( x  e.  dom  F  -> 
( F `  x
)  C_  B )
24 ndmfv 5818 . . . . . . . . . 10  |-  ( -.  x  e.  dom  F  ->  ( F `  x
)  =  (/) )
25 0ss 3769 . . . . . . . . . 10  |-  (/)  C_  B
2624, 25syl6eqss 3509 . . . . . . . . 9  |-  ( -.  x  e.  dom  F  ->  ( F `  x
)  C_  B )
2723, 26pm2.61i 164 . . . . . . . 8  |-  ( F `
 x )  C_  B
28 rsp 2888 . . . . . . . . 9  |-  ( A. x  e.  A  B  C_  C  ->  ( x  e.  A  ->  B  C_  C ) )
2928impcom 430 . . . . . . . 8  |-  ( ( x  e.  A  /\  A. x  e.  A  B  C_  C )  ->  B  C_  C )
3027, 29syl5ss 3470 . . . . . . 7  |-  ( ( x  e.  A  /\  A. x  e.  A  B  C_  C )  ->  ( F `  x )  C_  C )
3130ex 434 . . . . . 6  |-  ( x  e.  A  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  x )  C_  C ) )
327, 14, 17, 31vtoclgaf 3135 . . . . 5  |-  ( y  e.  A  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  y )  C_  C ) )
336, 32vtoclga 3136 . . . 4  |-  ( D  e.  A  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  D )  C_  C ) )
3433impcom 430 . . 3  |-  ( ( A. x  e.  A  B  C_  C  /\  D  e.  A )  ->  ( F `  D )  C_  C )
353, 34sylan2 474 . 2  |-  ( ( A. x  e.  A  B  C_  C  /\  D  e.  dom  F )  -> 
( F `  D
)  C_  C )
36 ndmfv 5818 . . . 4  |-  ( -.  D  e.  dom  F  ->  ( F `  D
)  =  (/) )
3736adantl 466 . . 3  |-  ( ( A. x  e.  A  B  C_  C  /\  -.  D  e.  dom  F )  ->  ( F `  D )  =  (/) )
38 0ss 3769 . . 3  |-  (/)  C_  C
3937, 38syl6eqss 3509 . 2  |-  ( ( A. x  e.  A  B  C_  C  /\  -.  D  e.  dom  F )  ->  ( F `  D )  C_  C
)
4035, 39pm2.61dan 789 1  |-  ( A. x  e.  A  B  C_  C  ->  ( F `  D )  C_  C
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2796   _Vcvv 3072    C_ wss 3431   (/)c0 3740    |-> cmpt 4453   dom cdm 4943   ` cfv 5521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fv 5529
This theorem is referenced by:  relmptopab  6413  ovmptss  6759
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