Theorem ovima0 7600

Description: An operation value is a member of the image plus null. (Contributed by Thierry Arnoux, 25-Jun-2019.)

Assertion

Ref	Expression
ovima0	⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}))

Proof of Theorem ovima0

Step	Hyp	Ref	Expression
1		simpr 484	. . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) = ∅)
2		ssun2 4173	. . . 4 ⊢ {∅} ⊆ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})
3		0ex 5307	. . . . 5 ⊢ ∅ ∈ V
4	3	snid 4665	. . . 4 ⊢ ∅ ∈ {∅}
5	2, 4	sselii 3977	. . 3 ⊢ ∅ ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})
6	1, 5	eqeltrdi 2837	. 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}))
7		ssun1 4172	. . 3 ⊢ (𝑅 “ (𝐴 × 𝐵)) ⊆ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})
8		df-ov 7423	. . . 4 ⊢ (𝑋𝑅𝑌) = (𝑅‘⟨𝑋, 𝑌⟩)
9		opelxpi 5715	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
10	8	eqeq1i 2733	. . . . . . 7 ⊢ ((𝑋𝑅𝑌) = ∅ ↔ (𝑅‘⟨𝑋, 𝑌⟩) = ∅)
11	10	notbii 320	. . . . . 6 ⊢ (¬ (𝑋𝑅𝑌) = ∅ ↔ ¬ (𝑅‘⟨𝑋, 𝑌⟩) = ∅)
12	11	biimpi 215	. . . . 5 ⊢ (¬ (𝑋𝑅𝑌) = ∅ → ¬ (𝑅‘⟨𝑋, 𝑌⟩) = ∅)
13		eliman0 6937	. . . . 5 ⊢ ((⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵) ∧ ¬ (𝑅‘⟨𝑋, 𝑌⟩) = ∅) → (𝑅‘⟨𝑋, 𝑌⟩) ∈ (𝑅 “ (𝐴 × 𝐵)))
14	9, 12, 13	syl2an 595	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑅‘⟨𝑋, 𝑌⟩) ∈ (𝑅 “ (𝐴 × 𝐵)))
15	8, 14	eqeltrid 2833	. . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ (𝑅 “ (𝐴 × 𝐵)))
16	7, 15	sselid 3978	. 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋𝑅𝑌) = ∅) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}))
17	6, 16	pm2.61dan 812	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099   ∪ cun 3945  ∅c0 4323  {csn 4629  ⟨cop 4635   × cxp 5676   “ cima 5681  ‘cfv 6548  (class class class)co 7420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fv 6556  df-ov 7423

This theorem is referenced by:  legval  28401