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Theorem nssdmovg 7311
 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) ∧ ¬ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) = ∅)

Proof of Theorem nssdmovg
StepHypRef Expression
1 df-ov 7138 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 ssel2 3910 . . . . 5 ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆))
3 opelxp 5555 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) ↔ (𝐴𝑅𝐵𝑆))
42, 3sylib 221 . . . 4 ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → (𝐴𝑅𝐵𝑆))
54stoic1a 1774 . . 3 ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → ¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6 ndmfv 6675 . . 3 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
75, 6syl 17 . 2 ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
81, 7syl5eq 2845 1 ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3881  ∅c0 4243  ⟨cop 4531   × cxp 5517  dom cdm 5519  ‘cfv 6324  (class class class)co 7135 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-dm 5529  df-iota 6283  df-fv 6332  df-ov 7138 This theorem is referenced by:  mpondm0  7367
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