Theorem elovmpt3imp 7404

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 4302	. 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 7388	. . . 4 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = ∅)
5		fveq1 6671	. . . . 5 ⊢ ((𝑋𝑂𝑌) = ∅ → ((𝑋𝑂𝑌)‘𝑍) = (∅‘𝑍))
6		0fv 6711	. . . . 5 ⊢ (∅‘𝑍) = ∅
7	5, 6	syl6eq 2874	. . . 4 ⊢ ((𝑋𝑂𝑌) = ∅ → ((𝑋𝑂𝑌)‘𝑍) = ∅)
8		eqneqall 3029	. . . 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 184	. 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 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  Vcvv 3496  ∅c0 4293   ↦ cmpt 5148  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-xp 5563  df-dm 5567  df-iota 6316  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163

This theorem is referenced by:  elovmpt3rab1  7407  elovmptnn0wrd  13913