Theorem elovmpt3imp 7526

Description: If the value of a function which is the result of an operation defined by the maps-to notation is not empty, the operands must be sets. Remark: a function which is the result of an operation can be regared as operation with 3 operands - therefore the abbreviation "mpt3" is used in the label. (Contributed by AV, 16-May-2019.)

Hypothesis

Ref	Expression
elovmpt3imp.o	⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧 ∈ 𝑀 ↦ 𝐵))

Assertion

Ref	Expression
elovmpt3imp	⊢ (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → (𝑋 ∈ V ∧ 𝑌 ∈ V))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)   𝐵(𝑥,𝑦,𝑧)   𝑀(𝑥,𝑦,𝑧)   𝑂(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦,𝑧)   𝑌(𝑥,𝑦,𝑧)   𝑍(𝑥,𝑦,𝑧)

Proof of Theorem elovmpt3imp

Step	Hyp	Ref	Expression
1		ne0i 4268	. 2 ⊢ (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → ((𝑋𝑂𝑌)‘𝑍) ≠ ∅)
2		ax-1 6	. . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (((𝑋𝑂𝑌)‘𝑍) ≠ ∅ → (𝑋 ∈ V ∧ 𝑌 ∈ V)))
3		elovmpt3imp.o	. . . . 5 ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧 ∈ 𝑀 ↦ 𝐵))
4	3	mpondm0 7510	. . . 4 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = ∅)
5		fveq1 6773	. . . . 5 ⊢ ((𝑋𝑂𝑌) = ∅ → ((𝑋𝑂𝑌)‘𝑍) = (∅‘𝑍))
6		0fv 6813	. . . . 5 ⊢ (∅‘𝑍) = ∅
7	5, 6	eqtrdi 2794	. . . 4 ⊢ ((𝑋𝑂𝑌) = ∅ → ((𝑋𝑂𝑌)‘𝑍) = ∅)
8		eqneqall 2954	. . . 4 ⊢ (((𝑋𝑂𝑌)‘𝑍) = ∅ → (((𝑋𝑂𝑌)‘𝑍) ≠ ∅ → (𝑋 ∈ V ∧ 𝑌 ∈ V)))
9	4, 7, 8	3syl 18	. . 3 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (((𝑋𝑂𝑌)‘𝑍) ≠ ∅ → (𝑋 ∈ V ∧ 𝑌 ∈ V)))
10	2, 9	pm2.61i 182	. 2 ⊢ (((𝑋𝑂𝑌)‘𝑍) ≠ ∅ → (𝑋 ∈ V ∧ 𝑌 ∈ V))
11	1, 10	syl 17	1 ⊢ (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → (𝑋 ∈ V ∧ 𝑌 ∈ V))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  Vcvv 3432  ∅c0 4256   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-dm 5599  df-iota 6391  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280

This theorem is referenced by:  elovmpt3rab1  7529  elovmptnn0wrd  14262