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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 2526 . . . 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 2555 1  |-  ( ( F `  A )  e.  S  ->  A  e.  S )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1395    e. wcel 1819   (/)c0 3793   dom cdm 5008   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586  ax-pow 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-dm 5018  df-iota 5557  df-fv 5602
This theorem is referenced by:  lterpq  9365  ltrnq  9374  reclem2pr  9443  msrrcl  29100
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