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Theorem nssdmovg 6776
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 6613 . 2 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 ssel2 3582 . . . . 5 ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆))
3 opelxp 5111 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) ↔ (𝐴𝑅𝐵𝑆))
42, 3sylib 208 . . . 4 ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹) → (𝐴𝑅𝐵𝑆))
54stoic1a 1694 . . 3 ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → ¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6 ndmfv 6180 . . 3 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
75, 6syl 17 . 2 ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
81, 7syl5eq 2667 1 ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  wss 3559  c0 3896  cop 4159   × cxp 5077  dom cdm 5079  cfv 5852  (class class class)co 6610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-xp 5085  df-dm 5089  df-iota 5815  df-fv 5860  df-ov 6613
This theorem is referenced by:  mpt2ndm0  6835
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