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Theorem ndmfvrcl 5553
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 5552 . . . . 5  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
32eleq1d 2349 . . . 4  |-  ( -.  A  e.  dom  F  ->  ( ( F `  A )  e.  S  <->  (/)  e.  S ) )
41, 3mtbiri 294 . . 3  |-  ( -.  A  e.  dom  F  ->  -.  ( F `  A )  e.  S
)
54con4i 122 . 2  |-  ( ( F `  A )  e.  S  ->  A  e.  dom  F )
6 ndmfvrcl.1 . 2  |-  dom  F  =  S
75, 6syl6eleq 2373 1  |-  ( ( F `  A )  e.  S  ->  A  e.  S )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684   (/)c0 3455   dom cdm 4689   ` cfv 5255
This theorem is referenced by:  lterpq  8594  ltrnq  8603  reclem2pr  8672
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-nul 4149  ax-pow 4188
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-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-dm 4699  df-iota 5219  df-fv 5263
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