MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvmptss2 Unicode version

Theorem fvmptss2 5702
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 2424 . . . 4  |-  ( x  =  D  ->  ( B  e.  _V  <->  C  e.  _V ) )
3 fvmptn.2 . . . . 5  |-  F  =  ( x  e.  A  |->  B )
43dmmpt 5250 . . . 4  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
52, 4elrab2 3001 . . 3  |-  ( D  e.  dom  F  <->  ( D  e.  A  /\  C  e. 
_V ) )
61, 3fvmptg 5683 . . . 4  |-  ( ( D  e.  A  /\  C  e.  _V )  ->  ( F `  D
)  =  C )
7 eqimss 3306 . . . 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 5635 . . 3  |-  ( -.  D  e.  dom  F  ->  ( F `  D
)  =  (/) )
11 0ss 3559 . . . 4  |-  (/)  C_  C
1211a1i 10 . . 3  |-  ( -.  D  e.  dom  F  -> 
(/)  C_  C )
1310, 12eqsstrd 3288 . 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 1642    e. wcel 1710   _Vcvv 2864    C_ wss 3228   (/)c0 3531    e. cmpt 4158   dom cdm 4771   ` cfv 5337
This theorem is referenced by:  cvmsi  24200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fv 5345
  Copyright terms: Public domain W3C validator