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Theorem ovima0 6855
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 3810 . . . 4 {∅} ⊆ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})
3 0ex 4823 . . . . 5 ∅ ∈ V
43snid 4241 . . . 4 ∅ ∈ {∅}
52, 4sselii 3633 . . 3 ∅ ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})
61, 5syl6eqel 2738 . 2 (((𝑋𝐴𝑌𝐵) ∧ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}))
7 ssun1 3809 . . 3 (𝑅 “ (𝐴 × 𝐵)) ⊆ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})
8 df-ov 6693 . . . 4 (𝑋𝑅𝑌) = (𝑅‘⟨𝑋, 𝑌⟩)
9 opelxpi 5182 . . . . 5 ((𝑋𝐴𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
108eqeq1i 2656 . . . . . . 7 ((𝑋𝑅𝑌) = ∅ ↔ (𝑅‘⟨𝑋, 𝑌⟩) = ∅)
1110notbii 309 . . . . . 6 (¬ (𝑋𝑅𝑌) = ∅ ↔ ¬ (𝑅‘⟨𝑋, 𝑌⟩) = ∅)
1211biimpi 206 . . . . 5 (¬ (𝑋𝑅𝑌) = ∅ → ¬ (𝑅‘⟨𝑋, 𝑌⟩) = ∅)
13 eliman0 6261 . . . . 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 2734 . . 3 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ (𝑅 “ (𝐴 × 𝐵)))
167, 15sseldi 3634 . 2 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}))
176, 16pm2.61dan 849 1 ((𝑋𝐴𝑌𝐵) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  cun 3605  c0 3948  {csn 4210  cop 4216   × cxp 5141  cima 5146  cfv 5926  (class class class)co 6690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fv 5934  df-ov 6693
This theorem is referenced by:  legval  25524
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