Theorem elovmpt3imp 7624

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 4281	. 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 7607	. . . 4 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = ∅)
5		fveq1 6839	. . . . 5 ⊢ ((𝑋𝑂𝑌) = ∅ → ((𝑋𝑂𝑌)‘𝑍) = (∅‘𝑍))
6		0fv 6881	. . . . 5 ⊢ (∅‘𝑍) = ∅
7	5, 6	eqtrdi 2787	. . . 4 ⊢ ((𝑋𝑂𝑌) = ∅ → ((𝑋𝑂𝑌)‘𝑍) = ∅)
8		eqneqall 2943	. . . 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 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  Vcvv 3429  ∅c0 4273   ↦ cmpt 5166  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-xp 5637  df-dm 5641  df-iota 6454  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372

This theorem is referenced by:  elovmpt3rab1  7627  elovmptnn0wrd  14521