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Theorem fvmptss2 5791
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 2478 . . . 4  |-  ( x  =  D  ->  ( B  e.  _V  <->  C  e.  _V ) )
3 fvmptn.2 . . . . 5  |-  F  =  ( x  e.  A  |->  B )
43dmmpt 5332 . . . 4  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
52, 4elrab2 3062 . . 3  |-  ( D  e.  dom  F  <->  ( D  e.  A  /\  C  e. 
_V ) )
61, 3fvmptg 5771 . . . 4  |-  ( ( D  e.  A  /\  C  e.  _V )  ->  ( F `  D
)  =  C )
7 eqimss 3368 . . . 4  |-  ( ( F `  D )  =  C  ->  ( F `  D )  C_  C )
86, 7syl 16 . . 3  |-  ( ( D  e.  A  /\  C  e.  _V )  ->  ( F `  D
)  C_  C )
95, 8sylbi 188 . 2  |-  ( D  e.  dom  F  -> 
( F `  D
)  C_  C )
10 ndmfv 5722 . . 3  |-  ( -.  D  e.  dom  F  ->  ( F `  D
)  =  (/) )
11 0ss 3624 . . 3  |-  (/)  C_  C
1210, 11syl6eqss 3366 . 2  |-  ( -.  D  e.  dom  F  ->  ( F `  D
)  C_  C )
139, 12pm2.61i 158 1  |-  ( F `
 D )  C_  C
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2924    C_ wss 3288   (/)c0 3596    e. cmpt 4234   dom cdm 4845   ` cfv 5421
This theorem is referenced by:  cvmsi  24913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fv 5429
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