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Theorem fvmptss2 5619
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 2349 . . . 4  |-  ( x  =  D  ->  ( B  e.  _V  <->  C  e.  _V ) )
3 fvmptn.2 . . . . 5  |-  F  =  ( x  e.  A  |->  B )
43dmmpt 5168 . . . 4  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
52, 4elrab2 2925 . . 3  |-  ( D  e.  dom  F  <->  ( D  e.  A  /\  C  e. 
_V ) )
61, 3fvmptg 5600 . . . 4  |-  ( ( D  e.  A  /\  C  e.  _V )  ->  ( F `  D
)  =  C )
7 eqimss 3230 . . . 4  |-  ( ( F `  D )  =  C  ->  ( F `  D )  C_  C )
86, 7syl 15 . . 3  |-  ( ( D  e.  A  /\  C  e.  _V )  ->  ( F `  D
)  C_  C )
95, 8sylbi 187 . 2  |-  ( D  e.  dom  F  -> 
( F `  D
)  C_  C )
10 ndmfv 5552 . . 3  |-  ( -.  D  e.  dom  F  ->  ( F `  D
)  =  (/) )
11 0ss 3483 . . . 4  |-  (/)  C_  C
1211a1i 10 . . 3  |-  ( -.  D  e.  dom  F  -> 
(/)  C_  C )
1310, 12eqsstrd 3212 . 2  |-  ( -.  D  e.  dom  F  ->  ( F `  D
)  C_  C )
149, 13pm2.61i 156 1  |-  ( F `
 D )  C_  C
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   (/)c0 3455    e. cmpt 4077   dom cdm 4689   ` cfv 5255
This theorem is referenced by:  cvmsi  23796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263
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