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Theorem fvmptss2 5827
Description: A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.)
Hypotheses
Ref Expression
fvmptn.1  |-  ( x  =  D  ->  B  =  C )
fvmptn.2  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
fvmptss2  |-  ( F `
 D )  C_  C
Distinct variable groups:    x, A    x, C    x, D
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem fvmptss2
StepHypRef Expression
1 fvmptn.1 . . . . 5  |-  ( x  =  D  ->  B  =  C )
21eleq1d 2504 . . . 4  |-  ( x  =  D  ->  ( B  e.  _V  <->  C  e.  _V ) )
3 fvmptn.2 . . . . 5  |-  F  =  ( x  e.  A  |->  B )
43dmmpt 5368 . . . 4  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
52, 4elrab2 3096 . . 3  |-  ( D  e.  dom  F  <->  ( D  e.  A  /\  C  e. 
_V ) )
61, 3fvmptg 5807 . . . 4  |-  ( ( D  e.  A  /\  C  e.  _V )  ->  ( F `  D
)  =  C )
7 eqimss 3402 . . . 4  |-  ( ( F `  D )  =  C  ->  ( F `  D )  C_  C )
86, 7syl 16 . . 3  |-  ( ( D  e.  A  /\  C  e.  _V )  ->  ( F `  D
)  C_  C )
95, 8sylbi 189 . 2  |-  ( D  e.  dom  F  -> 
( F `  D
)  C_  C )
10 ndmfv 5758 . . 3  |-  ( -.  D  e.  dom  F  ->  ( F `  D
)  =  (/) )
11 0ss 3658 . . 3  |-  (/)  C_  C
1210, 11syl6eqss 3400 . 2  |-  ( -.  D  e.  dom  F  ->  ( F `  D
)  C_  C )
139, 12pm2.61i 159 1  |-  ( F `
 D )  C_  C
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   (/)c0 3630    e. cmpt 4269   dom cdm 4881   ` cfv 5457
This theorem is referenced by:  cvmsi  24957
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-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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