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Theorem ndmfvrcl 6942
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 6941 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
32eleq1d 2826 . . . 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 2851 1 ((𝐹𝐴) ∈ 𝑆𝐴𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2108  c0 4333  dom cdm 5685  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-dm 5695  df-iota 6514  df-fv 6569
This theorem is referenced by:  lterpq  11010  ltrnq  11019  reclem2pr  11088  msrrcl  35548
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