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Theorem ndmfvrcl 5748
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 5747 . . . . 5  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
32eleq1d 2501 . . . 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 2525 1  |-  ( ( F `  A )  e.  S  ->  A  e.  S )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725   (/)c0 3620   dom cdm 4870   ` cfv 5446
This theorem is referenced by:  lterpq  8839  ltrnq  8848  reclem2pr  8917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330  ax-pow 4369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-dm 4880  df-iota 5410  df-fv 5454
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