Theorem nssdmovg 7432

Description: The value of an operation outside its domain. (Contributed by Alexander van der Vekens, 7-Sep-2017.)

Assertion

Ref	Expression
nssdmovg	⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅)

Proof of Theorem nssdmovg

Step	Hyp	Ref	Expression
1		df-ov 7258	. 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		ssel2 3912	. . . . 5 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆))
3		opelxp 5616	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆))
4	2, 3	sylib 217	. . . 4 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆))
5	4	stoic1a 1776	. . 3 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6		ndmfv 6786	. . 3 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
7	5, 6	syl 17	. 2 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
8	1, 7	eqtrid 2790	1 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ⊆ wss 3883  ∅c0 4253  ⟨cop 4564   × cxp 5578  dom cdm 5580  ‘cfv 6418  (class class class)co 7255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-dm 5590  df-iota 6376  df-fv 6426  df-ov 7258

This theorem is referenced by:  mpondm0  7488