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Theorem fvmptss 5816
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 5369 . . . 4  |-  dom  F  C_  A
32sseli 3346 . . 3  |-  ( D  e.  dom  F  ->  D  e.  A )
4 fveq2 5731 . . . . . . 7  |-  ( y  =  D  ->  ( F `  y )  =  ( F `  D ) )
54sseq1d 3377 . . . . . 6  |-  ( y  =  D  ->  (
( F `  y
)  C_  C  <->  ( F `  D )  C_  C
) )
65imbi2d 309 . . . . 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 2574 . . . . . 6  |-  F/_ x
y
8 nfra1 2758 . . . . . . 7  |-  F/ x A. x  e.  A  B  C_  C
9 nfmpt1 4301 . . . . . . . . . 10  |-  F/_ x
( x  e.  A  |->  B )
101, 9nfcxfr 2571 . . . . . . . . 9  |-  F/_ x F
1110, 7nffv 5738 . . . . . . . 8  |-  F/_ x
( F `  y
)
12 nfcv 2574 . . . . . . . 8  |-  F/_ x C
1311, 12nfss 3343 . . . . . . 7  |-  F/ x
( F `  y
)  C_  C
148, 13nfim 1833 . . . . . 6  |-  F/ x
( A. x  e.  A  B  C_  C  ->  ( F `  y
)  C_  C )
15 fveq2 5731 . . . . . . . 8  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
1615sseq1d 3377 . . . . . . 7  |-  ( x  =  y  ->  (
( F `  x
)  C_  C  <->  ( F `  y )  C_  C
) )
1716imbi2d 309 . . . . . 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 5368 . . . . . . . . . . 11  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
1918rabeq2i 2955 . . . . . . . . . 10  |-  ( x  e.  dom  F  <->  ( x  e.  A  /\  B  e. 
_V ) )
201fvmpt2 5815 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  B  e.  _V )  ->  ( F `  x
)  =  B )
21 eqimss 3402 . . . . . . . . . . 11  |-  ( ( F `  x )  =  B  ->  ( F `  x )  C_  B )
2220, 21syl 16 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  B  e.  _V )  ->  ( F `  x
)  C_  B )
2319, 22sylbi 189 . . . . . . . . 9  |-  ( x  e.  dom  F  -> 
( F `  x
)  C_  B )
24 ndmfv 5758 . . . . . . . . . 10  |-  ( -.  x  e.  dom  F  ->  ( F `  x
)  =  (/) )
25 0ss 3658 . . . . . . . . . 10  |-  (/)  C_  B
2624, 25syl6eqss 3400 . . . . . . . . 9  |-  ( -.  x  e.  dom  F  ->  ( F `  x
)  C_  B )
2723, 26pm2.61i 159 . . . . . . . 8  |-  ( F `
 x )  C_  B
28 rsp 2768 . . . . . . . . 9  |-  ( A. x  e.  A  B  C_  C  ->  ( x  e.  A  ->  B  C_  C ) )
2928impcom 421 . . . . . . . 8  |-  ( ( x  e.  A  /\  A. x  e.  A  B  C_  C )  ->  B  C_  C )
3027, 29syl5ss 3361 . . . . . . 7  |-  ( ( x  e.  A  /\  A. x  e.  A  B  C_  C )  ->  ( F `  x )  C_  C )
3130ex 425 . . . . . 6  |-  ( x  e.  A  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  x )  C_  C ) )
327, 14, 17, 31vtoclgaf 3018 . . . . 5  |-  ( y  e.  A  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  y )  C_  C ) )
336, 32vtoclga 3019 . . . 4  |-  ( D  e.  A  ->  ( A. x  e.  A  B  C_  C  ->  ( F `  D )  C_  C ) )
3433impcom 421 . . 3  |-  ( ( A. x  e.  A  B  C_  C  /\  D  e.  A )  ->  ( F `  D )  C_  C )
353, 34sylan2 462 . 2  |-  ( ( A. x  e.  A  B  C_  C  /\  D  e.  dom  F )  -> 
( F `  D
)  C_  C )
36 ndmfv 5758 . . . 4  |-  ( -.  D  e.  dom  F  ->  ( F `  D
)  =  (/) )
3736adantl 454 . . 3  |-  ( ( A. x  e.  A  B  C_  C  /\  -.  D  e.  dom  F )  ->  ( F `  D )  =  (/) )
38 0ss 3658 . . 3  |-  (/)  C_  C
3937, 38syl6eqss 3400 . 2  |-  ( ( A. x  e.  A  B  C_  C  /\  -.  D  e.  dom  F )  ->  ( F `  D )  C_  C
)
4035, 39pm2.61dan 768 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 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    C_ wss 3322   (/)c0 3630    e. cmpt 4269   dom cdm 4881   ` cfv 5457
This theorem is referenced by:  relmptopab  6295  ovmptss  6431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fv 5465
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