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Theorem fvmptss2 5967
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 2536 . . . 4  |-  ( x  =  D  ->  ( B  e.  _V  <->  C  e.  _V ) )
3 fvmptn.2 . . . . 5  |-  F  =  ( x  e.  A  |->  B )
43dmmpt 5500 . . . 4  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
52, 4elrab2 3263 . . 3  |-  ( D  e.  dom  F  <->  ( D  e.  A  /\  C  e. 
_V ) )
61, 3fvmptg 5946 . . . 4  |-  ( ( D  e.  A  /\  C  e.  _V )  ->  ( F `  D
)  =  C )
7 eqimss 3556 . . . 4  |-  ( ( F `  D )  =  C  ->  ( F `  D )  C_  C )
86, 7syl 16 . . 3  |-  ( ( D  e.  A  /\  C  e.  _V )  ->  ( F `  D
)  C_  C )
95, 8sylbi 195 . 2  |-  ( D  e.  dom  F  -> 
( F `  D
)  C_  C )
10 ndmfv 5888 . . 3  |-  ( -.  D  e.  dom  F  ->  ( F `  D
)  =  (/) )
11 0ss 3814 . . 3  |-  (/)  C_  C
1210, 11syl6eqss 3554 . 2  |-  ( -.  D  e.  dom  F  ->  ( F `  D
)  C_  C )
139, 12pm2.61i 164 1  |-  ( F `
 D )  C_  C
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   (/)c0 3785    |-> cmpt 4505   dom cdm 4999   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fv 5594
This theorem is referenced by:  cvmsi  28350
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