Theorem ndmfvrcl 6916

Description: Reverse closure law for function with the empty set not in its domain (if 𝑅 = 𝑆). (Contributed by NM, 26-Apr-1996.) The class containing the function value does not have to be the domain. (Revised by Zhi Wang, 10-Nov-2025.)

Hypotheses

Ref	Expression
ndmfvrcl.1	⊢ dom 𝐹 = 𝑆
ndmfvrcl.2	⊢ ¬ ∅ ∈ 𝑅

Assertion

Ref	Expression
ndmfvrcl	⊢ ((𝐹‘𝐴) ∈ 𝑅 → 𝐴 ∈ 𝑆)

Proof of Theorem ndmfvrcl

Step	Hyp	Ref	Expression
1		ndmfvrcl.2	. . . 4 ⊢ ¬ ∅ ∈ 𝑅
2		ndmfv 6915	. . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
3	2	eleq1d 2848	. . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ((𝐹‘𝐴) ∈ 𝑅 ↔ ∅ ∈ 𝑅))
4	1, 3	mtbiri 330	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ¬ (𝐹‘𝐴) ∈ 𝑅)
5	4	con4i 115	. 2 ⊢ ((𝐹‘𝐴) ∈ 𝑅 → 𝐴 ∈ dom 𝐹)
6		ndmfvrcl.1	. 2 ⊢ dom 𝐹 = 𝑆
7	5, 6	eleqtrdi 2873	1 ⊢ ((𝐹‘𝐴) ∈ 𝑅 → 𝐴 ∈ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2143  ∅c0 4287  dom cdm 5663  ‘cfv 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-dm 5673  df-iota 6494  df-fv 6546

This theorem is referenced by:  lterpq  10956  ltrnq  10965  reclem2pr  11034  msrrcl  36016  idfurcl  49859