Theorem ndmfvrcl 6921

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 6920	. . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
3	2	eleq1d 2851	. . . 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 2876	1 ⊢ ((𝐹‘𝐴) ∈ 𝑅 → 𝐴 ∈ 𝑆)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2146  ∅c0 4289  dom cdm 5666  ‘cfv 6543

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-dm 5676  df-iota 6499  df-fv 6551

This theorem is used by:  lterpq  10973  ltrnq  10982  reclem2pr  11051  msrrcl  36056  idfurcl  49917