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Theorem fvmptss 5958
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 5331 . . . 4  |-  dom  F  C_  A
32sseli 3428 . . 3  |-  ( D  e.  dom  F  ->  D  e.  A )
4 fveq2 5865 . . . . . . 7  |-  ( y  =  D  ->  ( F `  y )  =  ( F `  D ) )
54sseq1d 3459 . . . . . 6  |-  ( y  =  D  ->  (
( F `  y
)  C_  C  <->  ( F `  D )  C_  C
) )
65imbi2d 318 . . . . 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 2592 . . . . . 6  |-  F/_ x
y
8 nfra1 2769 . . . . . . 7  |-  F/ x A. x  e.  A  B  C_  C
9 nfmpt1 4492 . . . . . . . . . 10  |-  F/_ x
( x  e.  A  |->  B )
101, 9nfcxfr 2590 . . . . . . . . 9  |-  F/_ x F
1110, 7nffv 5872 . . . . . . . 8  |-  F/_ x
( F `  y
)
12 nfcv 2592 . . . . . . . 8  |-  F/_ x C
1311, 12nfss 3425 . . . . . . 7  |-  F/ x
( F `  y
)  C_  C
148, 13nfim 2003 . . . . . 6  |-  F/ x
( A. x  e.  A  B  C_  C  ->  ( F `  y
)  C_  C )
15 fveq2 5865 . . . . . . . 8  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
1615sseq1d 3459 . . . . . . 7  |-  ( x  =  y  ->  (
( F `  x
)  C_  C  <->  ( F `  y )  C_  C
) )
1716imbi2d 318 . . . . . 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 5330 . . . . . . . . . . 11  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
1918rabeq2i 3042 . . . . . . . . . 10  |-  ( x  e.  dom  F  <->  ( x  e.  A  /\  B  e. 
_V ) )
201fvmpt2 5957 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  B  e.  _V )  ->  ( F `  x
)  =  B )
21 eqimss 3484 . . . . . . . . . . 11  |-  ( ( F `  x )  =  B  ->  ( F `  x )  C_  B )
2220, 21syl 17 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  B  e.  _V )  ->  ( F `  x
)  C_  B )
2319, 22sylbi 199 . . . . . . . . 9  |-  ( x  e.  dom  F  -> 
( F `  x
)  C_  B )
24 ndmfv 5889 . . . . . . . . . 10  |-  ( -.  x  e.  dom  F  ->  ( F `  x
)  =  (/) )
25 0ss 3763 . . . . . . . . . 10  |-  (/)  C_  B
2624, 25syl6eqss 3482 . . . . . . . . 9  |-  ( -.  x  e.  dom  F  ->  ( F `  x
)  C_  B )
2723, 26pm2.61i 168 . . . . . . . 8  |-  ( F `
 x )  C_  B
28 rsp 2754 . . . . . . . . 9  |-  ( A. x  e.  A  B  C_  C  ->  ( x  e.  A  ->  B  C_  C ) )
2928impcom 432 . . . . . . . 8  |-  ( ( x  e.  A  /\  A. x  e.  A  B  C_  C )  ->  B  C_  C )
3027, 29syl5ss 3443 . . . . . . 7  |-  ( ( x  e.  A  /\  A. x  e.  A  B  C_  C )  ->  ( F `  x )  C_  C )
3130ex 436 . . . . . 6  |-  ( x  e.  A  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  x )  C_  C ) )
327, 14, 17, 31vtoclgaf 3112 . . . . 5  |-  ( y  e.  A  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  y )  C_  C ) )
336, 32vtoclga 3113 . . . 4  |-  ( D  e.  A  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  D )  C_  C ) )
3433impcom 432 . . 3  |-  ( ( A. x  e.  A  B  C_  C  /\  D  e.  A )  ->  ( F `  D )  C_  C )
353, 34sylan2 477 . 2  |-  ( ( A. x  e.  A  B  C_  C  /\  D  e.  dom  F )  -> 
( F `  D
)  C_  C )
36 ndmfv 5889 . . . 4  |-  ( -.  D  e.  dom  F  ->  ( F `  D
)  =  (/) )
3736adantl 468 . . 3  |-  ( ( A. x  e.  A  B  C_  C  /\  -.  D  e.  dom  F )  ->  ( F `  D )  =  (/) )
38 0ss 3763 . . 3  |-  (/)  C_  C
3937, 38syl6eqss 3482 . 2  |-  ( ( A. x  e.  A  B  C_  C  /\  -.  D  e.  dom  F )  ->  ( F `  D )  C_  C
)
4035, 39pm2.61dan 800 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 371    = wceq 1444    e. wcel 1887   A.wral 2737   _Vcvv 3045    C_ wss 3404   (/)c0 3731    |-> cmpt 4461   dom cdm 4834   ` cfv 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fv 5590
This theorem is referenced by:  relmptopab  6517  ovmptss  6877
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