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 3978	. . . . 5 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆))
3		opelxp 5721	. . . . 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 1772	. . 3 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6		ndmfv 6941	. . 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 2789	1 ⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951  ∅c0 4333  ⟨cop 4632   × cxp 5683  dom cdm 5685  ‘cfv 6561  (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 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-dm 5695  df-iota 6514  df-fv 6569  df-ov 7434

This theorem is referenced by:  mpondm0  7673