Theorem ndmfvrcl 6897

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 6896	. . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
3	2	eleq1d 2814	. . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ((𝐹‘𝐴) ∈ 𝑅 ↔ ∅ ∈ 𝑅))
4	1, 3	mtbiri 327	. . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ¬ (𝐹‘𝐴) ∈ 𝑅)
5	4	con4i 114	. 2 ⊢ ((𝐹‘𝐴) ∈ 𝑅 → 𝐴 ∈ dom 𝐹)
6		ndmfvrcl.1	. 2 ⊢ dom 𝐹 = 𝑆
7	5, 6	eleqtrdi 2839	1 ⊢ ((𝐹‘𝐴) ∈ 𝑅 → 𝐴 ∈ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109  ∅c0 4299  dom cdm 5641  ‘cfv 6514

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-dm 5651  df-iota 6467  df-fv 6522

This theorem is referenced by:  lterpq  10930  ltrnq  10939  reclem2pr  11008  msrrcl  35537  idfurcl  49091