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Theorem ndmfvrcl 6938
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 ¬ ∅ ∈ 𝑆
Assertion
Ref Expression
ndmfvrcl ((𝐹𝐴) ∈ 𝑆𝐴𝑆)

Proof of Theorem ndmfvrcl
StepHypRef Expression
1 ndmfvrcl.2 . . . 4 ¬ ∅ ∈ 𝑆
2 ndmfv 6937 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
32eleq1d 2814 . . . 4 𝐴 ∈ dom 𝐹 → ((𝐹𝐴) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
41, 3mtbiri 326 . . 3 𝐴 ∈ dom 𝐹 → ¬ (𝐹𝐴) ∈ 𝑆)
54con4i 114 . 2 ((𝐹𝐴) ∈ 𝑆𝐴 ∈ dom 𝐹)
6 ndmfvrcl.1 . 2 dom 𝐹 = 𝑆
75, 6eleqtrdi 2839 1 ((𝐹𝐴) ∈ 𝑆𝐴𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1533  wcel 2098  c0 4326  dom cdm 5682  cfv 6553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-dm 5692  df-iota 6505  df-fv 6561
This theorem is referenced by:  lterpq  11001  ltrnq  11010  reclem2pr  11079  msrrcl  35186
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