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Theorem fvmptss 5609
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 5169 . . . 4  |-  dom  F  C_  A
32sseli 3176 . . 3  |-  ( D  e.  dom  F  ->  D  e.  A )
4 fveq2 5525 . . . . . . 7  |-  ( y  =  D  ->  ( F `  y )  =  ( F `  D ) )
54sseq1d 3205 . . . . . 6  |-  ( y  =  D  ->  (
( F `  y
)  C_  C  <->  ( F `  D )  C_  C
) )
65imbi2d 307 . . . . 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 2419 . . . . . 6  |-  F/_ x
y
8 nfra1 2593 . . . . . . 7  |-  F/ x A. x  e.  A  B  C_  C
9 nfmpt1 4109 . . . . . . . . . 10  |-  F/_ x
( x  e.  A  |->  B )
101, 9nfcxfr 2416 . . . . . . . . 9  |-  F/_ x F
1110, 7nffv 5532 . . . . . . . 8  |-  F/_ x
( F `  y
)
12 nfcv 2419 . . . . . . . 8  |-  F/_ x C
1311, 12nfss 3173 . . . . . . 7  |-  F/ x
( F `  y
)  C_  C
148, 13nfim 1769 . . . . . 6  |-  F/ x
( A. x  e.  A  B  C_  C  ->  ( F `  y
)  C_  C )
15 fveq2 5525 . . . . . . . 8  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
1615sseq1d 3205 . . . . . . 7  |-  ( x  =  y  ->  (
( F `  x
)  C_  C  <->  ( F `  y )  C_  C
) )
1716imbi2d 307 . . . . . 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 5168 . . . . . . . . . . 11  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
1918rabeq2i 2785 . . . . . . . . . 10  |-  ( x  e.  dom  F  <->  ( x  e.  A  /\  B  e. 
_V ) )
201fvmpt2 5608 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  B  e.  _V )  ->  ( F `  x
)  =  B )
21 eqimss 3230 . . . . . . . . . . 11  |-  ( ( F `  x )  =  B  ->  ( F `  x )  C_  B )
2220, 21syl 15 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  B  e.  _V )  ->  ( F `  x
)  C_  B )
2319, 22sylbi 187 . . . . . . . . 9  |-  ( x  e.  dom  F  -> 
( F `  x
)  C_  B )
24 ndmfv 5552 . . . . . . . . . 10  |-  ( -.  x  e.  dom  F  ->  ( F `  x
)  =  (/) )
25 0ss 3483 . . . . . . . . . . 11  |-  (/)  C_  B
2625a1i 10 . . . . . . . . . 10  |-  ( -.  x  e.  dom  F  -> 
(/)  C_  B )
2724, 26eqsstrd 3212 . . . . . . . . 9  |-  ( -.  x  e.  dom  F  ->  ( F `  x
)  C_  B )
2823, 27pm2.61i 156 . . . . . . . 8  |-  ( F `
 x )  C_  B
29 rsp 2603 . . . . . . . . 9  |-  ( A. x  e.  A  B  C_  C  ->  ( x  e.  A  ->  B  C_  C ) )
3029impcom 419 . . . . . . . 8  |-  ( ( x  e.  A  /\  A. x  e.  A  B  C_  C )  ->  B  C_  C )
3128, 30syl5ss 3190 . . . . . . 7  |-  ( ( x  e.  A  /\  A. x  e.  A  B  C_  C )  ->  ( F `  x )  C_  C )
3231ex 423 . . . . . 6  |-  ( x  e.  A  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  x )  C_  C ) )
337, 14, 17, 32vtoclgaf 2848 . . . . 5  |-  ( y  e.  A  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  y )  C_  C ) )
346, 33vtoclga 2849 . . . 4  |-  ( D  e.  A  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  D )  C_  C ) )
3534impcom 419 . . 3  |-  ( ( A. x  e.  A  B  C_  C  /\  D  e.  A )  ->  ( F `  D )  C_  C )
363, 35sylan2 460 . 2  |-  ( ( A. x  e.  A  B  C_  C  /\  D  e.  dom  F )  -> 
( F `  D
)  C_  C )
37 ndmfv 5552 . . . 4  |-  ( -.  D  e.  dom  F  ->  ( F `  D
)  =  (/) )
3837adantl 452 . . 3  |-  ( ( A. x  e.  A  B  C_  C  /\  -.  D  e.  dom  F )  ->  ( F `  D )  =  (/) )
39 0ss 3483 . . . 4  |-  (/)  C_  C
4039a1i 10 . . 3  |-  ( ( A. x  e.  A  B  C_  C  /\  -.  D  e.  dom  F )  ->  (/)  C_  C )
4138, 40eqsstrd 3212 . 2  |-  ( ( A. x  e.  A  B  C_  C  /\  -.  D  e.  dom  F )  ->  ( F `  D )  C_  C
)
4236, 41pm2.61dan 766 1  |-  ( A. x  e.  A  B  C_  C  ->  ( F `  D )  C_  C
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   (/)c0 3455    e. cmpt 4077   dom cdm 4689   ` cfv 5255
This theorem is referenced by:  relmptopab  6065  ovmptss  6200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263
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