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Theorem ovima0 6711
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 476 . . 3 (((𝑋𝐴𝑌𝐵) ∧ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) = ∅)
2 ssun2 3739 . . . 4 {∅} ⊆ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})
3 0ex 4718 . . . . 5 ∅ ∈ V
43snid 4155 . . . 4 ∅ ∈ {∅}
52, 4sselii 3565 . . 3 ∅ ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})
61, 5syl6eqel 2696 . 2 (((𝑋𝐴𝑌𝐵) ∧ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}))
7 ssun1 3738 . . 3 (𝑅 “ (𝐴 × 𝐵)) ⊆ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})
8 df-ov 6552 . . . 4 (𝑋𝑅𝑌) = (𝑅‘⟨𝑋, 𝑌⟩)
9 opelxpi 5072 . . . . 5 ((𝑋𝐴𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
108eqeq1i 2615 . . . . . . 7 ((𝑋𝑅𝑌) = ∅ ↔ (𝑅‘⟨𝑋, 𝑌⟩) = ∅)
1110notbii 309 . . . . . 6 (¬ (𝑋𝑅𝑌) = ∅ ↔ ¬ (𝑅‘⟨𝑋, 𝑌⟩) = ∅)
1211biimpi 205 . . . . 5 (¬ (𝑋𝑅𝑌) = ∅ → ¬ (𝑅‘⟨𝑋, 𝑌⟩) = ∅)
13 eliman0 6133 . . . . 5 ((⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵) ∧ ¬ (𝑅‘⟨𝑋, 𝑌⟩) = ∅) → (𝑅‘⟨𝑋, 𝑌⟩) ∈ (𝑅 “ (𝐴 × 𝐵)))
149, 12, 13syl2an 493 . . . 4 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑅‘⟨𝑋, 𝑌⟩) ∈ (𝑅 “ (𝐴 × 𝐵)))
158, 14syl5eqel 2692 . . 3 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ (𝑅 “ (𝐴 × 𝐵)))
167, 15sseldi 3566 . 2 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}))
176, 16pm2.61dan 828 1 ((𝑋𝐴𝑌𝐵) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1475  wcel 1977  cun 3538  c0 3874  {csn 4125  cop 4131   × cxp 5036  cima 5041  cfv 5804  (class class class)co 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fv 5812  df-ov 6552
This theorem is referenced by:  legval  25279
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