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Theorem ovima0 7525
Description: An operation value is a member of the image plus null. (Contributed by Thierry Arnoux, 25-Jun-2019.)
Assertion
Ref Expression
ovima0 ((𝑋𝐴𝑌𝐵) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}))

Proof of Theorem ovima0
StepHypRef Expression
1 simpr 484 . . 3 (((𝑋𝐴𝑌𝐵) ∧ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) = ∅)
2 ssun2 4129 . . . 4 {∅} ⊆ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})
3 0ex 5245 . . . . 5 ∅ ∈ V
43snid 4615 . . . 4 ∅ ∈ {∅}
52, 4sselii 3931 . . 3 ∅ ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})
61, 5eqeltrdi 2839 . 2 (((𝑋𝐴𝑌𝐵) ∧ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}))
7 ssun1 4128 . . 3 (𝑅 “ (𝐴 × 𝐵)) ⊆ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})
8 df-ov 7349 . . . 4 (𝑋𝑅𝑌) = (𝑅‘⟨𝑋, 𝑌⟩)
9 opelxpi 5653 . . . . 5 ((𝑋𝐴𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
108eqeq1i 2736 . . . . . . 7 ((𝑋𝑅𝑌) = ∅ ↔ (𝑅‘⟨𝑋, 𝑌⟩) = ∅)
1110notbii 320 . . . . . 6 (¬ (𝑋𝑅𝑌) = ∅ ↔ ¬ (𝑅‘⟨𝑋, 𝑌⟩) = ∅)
1211biimpi 216 . . . . 5 (¬ (𝑋𝑅𝑌) = ∅ → ¬ (𝑅‘⟨𝑋, 𝑌⟩) = ∅)
13 eliman0 6859 . . . . 5 ((⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵) ∧ ¬ (𝑅‘⟨𝑋, 𝑌⟩) = ∅) → (𝑅‘⟨𝑋, 𝑌⟩) ∈ (𝑅 “ (𝐴 × 𝐵)))
149, 12, 13syl2an 596 . . . 4 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑅‘⟨𝑋, 𝑌⟩) ∈ (𝑅 “ (𝐴 × 𝐵)))
158, 14eqeltrid 2835 . . 3 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ (𝑅 “ (𝐴 × 𝐵)))
167, 15sselid 3932 . 2 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}))
176, 16pm2.61dan 812 1 ((𝑋𝐴𝑌𝐵) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1541  wcel 2111  cun 3900  c0 4283  {csn 4576  cop 4582   × cxp 5614  cima 5619  cfv 6481  (class class class)co 7346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-xp 5622  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fv 6489  df-ov 7349
This theorem is referenced by:  legval  28563
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