Theorem nssdmovg 7615

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 7434	. 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		ssel2 3990	. . . . 5 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆))
3		opelxp 5725	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆))
4	2, 3	sylib 218	. . . 4 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆))
5	4	stoic1a 1769	. . 3 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6		ndmfv 6942	. . 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 2787	1 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ⊆ wss 3963  ∅c0 4339  ⟨cop 4637   × cxp 5687  dom cdm 5689  ‘cfv 6563  (class class class)co 7431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-dm 5699  df-iota 6516  df-fv 6571  df-ov 7434

This theorem is referenced by:  mpondm0  7673