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Theorem ndmfvrcl 6530
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 6529 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
32eleq1d 2851 . . . 4 𝐴 ∈ dom 𝐹 → ((𝐹𝐴) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
41, 3mtbiri 319 . . 3 𝐴 ∈ dom 𝐹 → ¬ (𝐹𝐴) ∈ 𝑆)
54con4i 114 . 2 ((𝐹𝐴) ∈ 𝑆𝐴 ∈ dom 𝐹)
6 ndmfvrcl.1 . 2 dom 𝐹 = 𝑆
75, 6syl6eleq 2877 1 ((𝐹𝐴) ∈ 𝑆𝐴𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1507  wcel 2050  c0 4179  dom cdm 5407  cfv 6188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751  ax-nul 5067  ax-pow 5119
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-dm 5417  df-iota 6152  df-fv 6196
This theorem is referenced by:  lterpq  10190  ltrnq  10199  reclem2pr  10268  msrrcl  32307
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