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Theorem ndmfvrcl 6943
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 6942 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
32eleq1d 2824 . . . 4 𝐴 ∈ dom 𝐹 → ((𝐹𝐴) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
41, 3mtbiri 327 . . 3 𝐴 ∈ dom 𝐹 → ¬ (𝐹𝐴) ∈ 𝑆)
54con4i 114 . 2 ((𝐹𝐴) ∈ 𝑆𝐴 ∈ dom 𝐹)
6 ndmfvrcl.1 . 2 dom 𝐹 = 𝑆
75, 6eleqtrdi 2849 1 ((𝐹𝐴) ∈ 𝑆𝐴𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106  c0 4339  dom cdm 5689  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-dm 5699  df-iota 6516  df-fv 6571
This theorem is referenced by:  lterpq  11008  ltrnq  11017  reclem2pr  11086  msrrcl  35528
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