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Theorem ovima0 7311
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 488 . . 3 (((𝑋𝐴𝑌𝐵) ∧ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) = ∅)
2 ssun2 4103 . . . 4 {∅} ⊆ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})
3 0ex 5178 . . . . 5 ∅ ∈ V
43snid 4564 . . . 4 ∅ ∈ {∅}
52, 4sselii 3915 . . 3 ∅ ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})
61, 5eqeltrdi 2901 . 2 (((𝑋𝐴𝑌𝐵) ∧ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}))
7 ssun1 4102 . . 3 (𝑅 “ (𝐴 × 𝐵)) ⊆ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})
8 df-ov 7142 . . . 4 (𝑋𝑅𝑌) = (𝑅‘⟨𝑋, 𝑌⟩)
9 opelxpi 5560 . . . . 5 ((𝑋𝐴𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
108eqeq1i 2806 . . . . . . 7 ((𝑋𝑅𝑌) = ∅ ↔ (𝑅‘⟨𝑋, 𝑌⟩) = ∅)
1110notbii 323 . . . . . 6 (¬ (𝑋𝑅𝑌) = ∅ ↔ ¬ (𝑅‘⟨𝑋, 𝑌⟩) = ∅)
1211biimpi 219 . . . . 5 (¬ (𝑋𝑅𝑌) = ∅ → ¬ (𝑅‘⟨𝑋, 𝑌⟩) = ∅)
13 eliman0 6684 . . . . 5 ((⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵) ∧ ¬ (𝑅‘⟨𝑋, 𝑌⟩) = ∅) → (𝑅‘⟨𝑋, 𝑌⟩) ∈ (𝑅 “ (𝐴 × 𝐵)))
149, 12, 13syl2an 598 . . . 4 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑅‘⟨𝑋, 𝑌⟩) ∈ (𝑅 “ (𝐴 × 𝐵)))
158, 14eqeltrid 2897 . . 3 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ (𝑅 “ (𝐴 × 𝐵)))
167, 15sseldi 3916 . 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 399   = wceq 1538  wcel 2112  cun 3882  c0 4246  {csn 4528  cop 4534   × cxp 5521  cima 5526  cfv 6328  (class class class)co 7139
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-xp 5529  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fv 6336  df-ov 7142
This theorem is referenced by:  legval  26381
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