Theorem relndmfv 5728

Description: The value of a relation outside its domain is the empty set. (Contributed by Jim Kingdon, 6-Aug-2026.)

Assertion

Ref	Expression
relndmfv	⊢ ((Rel 𝐹 ∧ ¬ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∅)

Proof of Theorem relndmfv

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relelfvdm 5727	. . . . . 6 ⊢ ((Rel 𝐹 ∧ 𝑥 ∈ (𝐹‘𝐴)) → 𝐴 ∈ dom 𝐹)
2	1	elexd 2835	. . . . 5 ⊢ ((Rel 𝐹 ∧ 𝑥 ∈ (𝐹‘𝐴)) → 𝐴 ∈ V)
3	2	adantlr 481	. . . 4 ⊢ (((Rel 𝐹 ∧ ¬ 𝐴 ∈ dom 𝐹) ∧ 𝑥 ∈ (𝐹‘𝐴)) → 𝐴 ∈ V)
4	3	ex 115	. . 3 ⊢ ((Rel 𝐹 ∧ ¬ 𝐴 ∈ dom 𝐹) → (𝑥 ∈ (𝐹‘𝐴) → 𝐴 ∈ V))
5		noel 3525	. . . . 5 ⊢ ¬ 𝑥 ∈ ∅
6	5	pm2.21i 655	. . . 4 ⊢ (𝑥 ∈ ∅ → 𝐴 ∈ V)
7	6	a1i 9	. . 3 ⊢ ((Rel 𝐹 ∧ ¬ 𝐴 ∈ dom 𝐹) → (𝑥 ∈ ∅ → 𝐴 ∈ V))
8		simpr 110	. . . . . 6 ⊢ (((Rel 𝐹 ∧ ¬ 𝐴 ∈ dom 𝐹) ∧ 𝐴 ∈ V) → 𝐴 ∈ V)
9		simplr 533	. . . . . 6 ⊢ (((Rel 𝐹 ∧ ¬ 𝐴 ∈ dom 𝐹) ∧ 𝐴 ∈ V) → ¬ 𝐴 ∈ dom 𝐹)
10		ndmfvg 5726	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∅)
11	8, 9, 10	syl2anc 415	. . . . 5 ⊢ (((Rel 𝐹 ∧ ¬ 𝐴 ∈ dom 𝐹) ∧ 𝐴 ∈ V) → (𝐹‘𝐴) = ∅)
12	11	eleq2d 2308	. . . 4 ⊢ (((Rel 𝐹 ∧ ¬ 𝐴 ∈ dom 𝐹) ∧ 𝐴 ∈ V) → (𝑥 ∈ (𝐹‘𝐴) ↔ 𝑥 ∈ ∅))
13	12	ex 115	. . 3 ⊢ ((Rel 𝐹 ∧ ¬ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ V → (𝑥 ∈ (𝐹‘𝐴) ↔ 𝑥 ∈ ∅)))
14	4, 7, 13	pm5.21ndd 717	. 2 ⊢ ((Rel 𝐹 ∧ ¬ 𝐴 ∈ dom 𝐹) → (𝑥 ∈ (𝐹‘𝐴) ↔ 𝑥 ∈ ∅))
15	14	eqrdv 2236	1 ⊢ ((Rel 𝐹 ∧ ¬ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  Vcvv 2821  ∅c0 3520  dom cdm 4774  Rel wrel 4779  ‘cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-dm 4784  df-iota 5337  df-fv 5385

This theorem is used by:  fvopab4ndm  5803