Theorem ndmfvrcl 6921

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	⊢ ((𝐹‘𝐴) ∈ 𝑆 → 𝐴 ∈ 𝑆)

Proof of Theorem ndmfvrcl

Step	Hyp	Ref	Expression
1		ndmfvrcl.2	. . . 4 ⊢ ¬ ∅ ∈ 𝑆
2		ndmfv 6920	. . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
3	2	eleq1d 2812	. . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ((𝐹‘𝐴) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
4	1, 3	mtbiri 327	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ¬ (𝐹‘𝐴) ∈ 𝑆)
5	4	con4i 114	. 2 ⊢ ((𝐹‘𝐴) ∈ 𝑆 → 𝐴 ∈ dom 𝐹)
6		ndmfvrcl.1	. 2 ⊢ dom 𝐹 = 𝑆
7	5, 6	eleqtrdi 2837	1 ⊢ ((𝐹‘𝐴) ∈ 𝑆 → 𝐴 ∈ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1533   ∈ wcel 2098  ∅c0 4317  dom cdm 5669  ‘cfv 6537

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 2163  ax-ext 2697  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-dm 5679  df-iota 6489  df-fv 6545

This theorem is referenced by:  lterpq  10967  ltrnq  10976  reclem2pr  11045  msrrcl  35062