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Theorem elovmpt3imp 7690
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 ↦ (𝑧𝑀𝐵))
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
StepHypRef Expression
1 ne0i 4341 . 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 ↦ (𝑧𝑀𝐵))
43mpondm0 7673 . . . 4 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = ∅)
5 fveq1 6905 . . . . 5 ((𝑋𝑂𝑌) = ∅ → ((𝑋𝑂𝑌)‘𝑍) = (∅‘𝑍))
6 0fv 6950 . . . . 5 (∅‘𝑍) = ∅
75, 6eqtrdi 2793 . . . 4 ((𝑋𝑂𝑌) = ∅ → ((𝑋𝑂𝑌)‘𝑍) = ∅)
8 eqneqall 2951 . . . 4 (((𝑋𝑂𝑌)‘𝑍) = ∅ → (((𝑋𝑂𝑌)‘𝑍) ≠ ∅ → (𝑋 ∈ V ∧ 𝑌 ∈ V)))
94, 7, 83syl 18 . . 3 (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (((𝑋𝑂𝑌)‘𝑍) ≠ ∅ → (𝑋 ∈ V ∧ 𝑌 ∈ V)))
102, 9pm2.61i 182 . 2 (((𝑋𝑂𝑌)‘𝑍) ≠ ∅ → (𝑋 ∈ V ∧ 𝑌 ∈ V))
111, 10syl 17 1 (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  c0 4333  cmpt 5225  cfv 6561  (class class class)co 7431  cmpo 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-dm 5695  df-iota 6514  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  elovmpt3rab1  7693  elovmptnn0wrd  14597
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