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Theorem ndmfvrcl 5698
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 5697 . . . . 5  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
32eleq1d 2455 . . . 4  |-  ( -.  A  e.  dom  F  ->  ( ( F `  A )  e.  S  <->  (/)  e.  S ) )
41, 3mtbiri 295 . . 3  |-  ( -.  A  e.  dom  F  ->  -.  ( F `  A )  e.  S
)
54con4i 124 . 2  |-  ( ( F `  A )  e.  S  ->  A  e.  dom  F )
6 ndmfvrcl.1 . 2  |-  dom  F  =  S
75, 6syl6eleq 2479 1  |-  ( ( F `  A )  e.  S  ->  A  e.  S )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1717   (/)c0 3573   dom cdm 4820   ` cfv 5396
This theorem is referenced by:  lterpq  8782  ltrnq  8791  reclem2pr  8860
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-nul 4281  ax-pow 4320
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-dm 4830  df-iota 5360  df-fv 5404
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