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

Theorem ndmfvrcl 5897
Description: Reverse closure law for function with the empty set not in its domain. (Contributed by NM, 26-Apr-1996.)
Hypotheses
Ref Expression
ndmfvrcl.1  |-  dom  F  =  S
ndmfvrcl.2  |-  -.  (/)  e.  S
Assertion
Ref Expression
ndmfvrcl  |-  ( ( F `  A )  e.  S  ->  A  e.  S )

Proof of Theorem ndmfvrcl
StepHypRef Expression
1 ndmfvrcl.2 . . . 4  |-  -.  (/)  e.  S
2 ndmfv 5896 . . . . 5  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
32eleq1d 2536 . . . 4  |-  ( -.  A  e.  dom  F  ->  ( ( F `  A )  e.  S  <->  (/)  e.  S ) )
41, 3mtbiri 303 . . 3  |-  ( -.  A  e.  dom  F  ->  -.  ( F `  A )  e.  S
)
54con4i 130 . 2  |-  ( ( F `  A )  e.  S  ->  A  e.  dom  F )
6 ndmfvrcl.1 . 2  |-  dom  F  =  S
75, 6syl6eleq 2565 1  |-  ( ( F `  A )  e.  S  ->  A  e.  S )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1379    e. wcel 1767   (/)c0 3790   dom cdm 5005   ` cfv 5594
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-nul 4582  ax-pow 4631
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-dm 5015  df-iota 5557  df-fv 5602
This theorem is referenced by:  lterpq  9360  ltrnq  9369  reclem2pr  9438  msrrcl  28728
  Copyright terms: Public domain W3C validator