Theorem ndmfvrcl 6694

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 6693	. . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
3	2	eleq1d 2896	. . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ((𝐹‘𝐴) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
4	1, 3	mtbiri 329	. . 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 2922	1 ⊢ ((𝐹‘𝐴) ∈ 𝑆 → 𝐴 ∈ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1536   ∈ wcel 2113  ∅c0 4284  dom cdm 5548  ‘cfv 6348

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-nul 5203  ax-pow 5259

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5060  df-dm 5558  df-iota 6307  df-fv 6356

This theorem is referenced by:  lterpq  10385  ltrnq  10394  reclem2pr  10463  msrrcl  32811