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Theorem fvmptss 5973
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
mptrcl.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 mptrcl.1 . . . . 5  |-  F  =  ( x  e.  A  |->  B )
21dmmptss 5338 . . . 4  |-  dom  F  C_  A
32sseli 3414 . . 3  |-  ( D  e.  dom  F  ->  D  e.  A )
4 fveq2 5879 . . . . . . 7  |-  ( y  =  D  ->  ( F `  y )  =  ( F `  D ) )
54sseq1d 3445 . . . . . 6  |-  ( y  =  D  ->  (
( F `  y
)  C_  C  <->  ( F `  D )  C_  C
) )
65imbi2d 323 . . . . 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 2612 . . . . . 6  |-  F/_ x
y
8 nfra1 2785 . . . . . . 7  |-  F/ x A. x  e.  A  B  C_  C
9 nfmpt1 4485 . . . . . . . . . 10  |-  F/_ x
( x  e.  A  |->  B )
101, 9nfcxfr 2610 . . . . . . . . 9  |-  F/_ x F
1110, 7nffv 5886 . . . . . . . 8  |-  F/_ x
( F `  y
)
12 nfcv 2612 . . . . . . . 8  |-  F/_ x C
1311, 12nfss 3411 . . . . . . 7  |-  F/ x
( F `  y
)  C_  C
148, 13nfim 2023 . . . . . 6  |-  F/ x
( A. x  e.  A  B  C_  C  ->  ( F `  y
)  C_  C )
15 fveq2 5879 . . . . . . . 8  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
1615sseq1d 3445 . . . . . . 7  |-  ( x  =  y  ->  (
( F `  x
)  C_  C  <->  ( F `  y )  C_  C
) )
1716imbi2d 323 . . . . . 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 5337 . . . . . . . . . . 11  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
1918rabeq2i 3028 . . . . . . . . . 10  |-  ( x  e.  dom  F  <->  ( x  e.  A  /\  B  e. 
_V ) )
201fvmpt2 5972 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  B  e.  _V )  ->  ( F `  x
)  =  B )
21 eqimss 3470 . . . . . . . . . . 11  |-  ( ( F `  x )  =  B  ->  ( F `  x )  C_  B )
2220, 21syl 17 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  B  e.  _V )  ->  ( F `  x
)  C_  B )
2319, 22sylbi 200 . . . . . . . . 9  |-  ( x  e.  dom  F  -> 
( F `  x
)  C_  B )
24 ndmfv 5903 . . . . . . . . . 10  |-  ( -.  x  e.  dom  F  ->  ( F `  x
)  =  (/) )
25 0ss 3766 . . . . . . . . . 10  |-  (/)  C_  B
2624, 25syl6eqss 3468 . . . . . . . . 9  |-  ( -.  x  e.  dom  F  ->  ( F `  x
)  C_  B )
2723, 26pm2.61i 169 . . . . . . . 8  |-  ( F `
 x )  C_  B
28 rsp 2773 . . . . . . . . 9  |-  ( A. x  e.  A  B  C_  C  ->  ( x  e.  A  ->  B  C_  C ) )
2928impcom 437 . . . . . . . 8  |-  ( ( x  e.  A  /\  A. x  e.  A  B  C_  C )  ->  B  C_  C )
3027, 29syl5ss 3429 . . . . . . 7  |-  ( ( x  e.  A  /\  A. x  e.  A  B  C_  C )  ->  ( F `  x )  C_  C )
3130ex 441 . . . . . 6  |-  ( x  e.  A  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  x )  C_  C ) )
327, 14, 17, 31vtoclgaf 3098 . . . . 5  |-  ( y  e.  A  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  y )  C_  C ) )
336, 32vtoclga 3099 . . . 4  |-  ( D  e.  A  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  D )  C_  C ) )
3433impcom 437 . . 3  |-  ( ( A. x  e.  A  B  C_  C  /\  D  e.  A )  ->  ( F `  D )  C_  C )
353, 34sylan2 482 . 2  |-  ( ( A. x  e.  A  B  C_  C  /\  D  e.  dom  F )  -> 
( F `  D
)  C_  C )
36 ndmfv 5903 . . . 4  |-  ( -.  D  e.  dom  F  ->  ( F `  D
)  =  (/) )
3736adantl 473 . . 3  |-  ( ( A. x  e.  A  B  C_  C  /\  -.  D  e.  dom  F )  ->  ( F `  D )  =  (/) )
38 0ss 3766 . . 3  |-  (/)  C_  C
3937, 38syl6eqss 3468 . 2  |-  ( ( A. x  e.  A  B  C_  C  /\  -.  D  e.  dom  F )  ->  ( F `  D )  C_  C
)
4035, 39pm2.61dan 808 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 376    = wceq 1452    e. wcel 1904   A.wral 2756   _Vcvv 3031    C_ wss 3390   (/)c0 3722    |-> cmpt 4454   dom cdm 4839   ` cfv 5589
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-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  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-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  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-iota 5553  df-fun 5591  df-fv 5597
This theorem is referenced by:  relmptopab  6536  ovmptss  6896
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