MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ndmfvrcl Structured version   Visualization version   GIF version

Theorem ndmfvrcl 6257
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 6256 . . . . 5 𝐴 ∈ dom 𝐹 → (𝐹𝐴) = ∅)
32eleq1d 2715 . . . 4 𝐴 ∈ dom 𝐹 → ((𝐹𝐴) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
41, 3mtbiri 316 . . 3 𝐴 ∈ dom 𝐹 → ¬ (𝐹𝐴) ∈ 𝑆)
54con4i 113 . 2 ((𝐹𝐴) ∈ 𝑆𝐴 ∈ dom 𝐹)
6 ndmfvrcl.1 . 2 dom 𝐹 = 𝑆
75, 6syl6eleq 2740 1 ((𝐹𝐴) ∈ 𝑆𝐴𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1523  wcel 2030  c0 3948  dom cdm 5143  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822  ax-pow 4873
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-dm 5153  df-iota 5889  df-fv 5934
This theorem is referenced by:  lterpq  9830  ltrnq  9839  reclem2pr  9908  msrrcl  31566
  Copyright terms: Public domain W3C validator