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Theorem ovmpt3rabdm 7617
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 and the argument of the function must be sets. (Contributed by AV, 16-May-2019.)
Hypotheses
Ref Expression
ovmpt3rab1.o 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧𝑀 ↦ {𝑎𝑁𝜑}))
ovmpt3rab1.m ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑀 = 𝐾)
ovmpt3rab1.n ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑁 = 𝐿)
Assertion
Ref Expression
ovmpt3rabdm (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → dom (𝑋𝑂𝑌) = 𝐾)
Distinct variable groups:   𝑥,𝐾,𝑦,𝑧   𝐿,𝑎,𝑥,𝑦   𝑁,𝑎   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦   𝑥,𝑈,𝑦   𝑋,𝑎,𝑥,𝑦,𝑧   𝑌,𝑎,𝑥,𝑦,𝑧   𝑧,𝐿   𝑧,𝑇   𝑧,𝑈   𝑧,𝑉   𝑧,𝑊
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑎)   𝑇(𝑥,𝑦,𝑎)   𝑈(𝑎)   𝐾(𝑎)   𝑀(𝑥,𝑦,𝑧,𝑎)   𝑁(𝑥,𝑦,𝑧)   𝑂(𝑥,𝑦,𝑧,𝑎)   𝑉(𝑎)   𝑊(𝑎)

Proof of Theorem ovmpt3rabdm
StepHypRef Expression
1 ovmpt3rab1.o . . . . 5 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧𝑀 ↦ {𝑎𝑁𝜑}))
2 ovmpt3rab1.m . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑀 = 𝐾)
3 ovmpt3rab1.n . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑁 = 𝐿)
4 sbceq1a 3755 . . . . . 6 (𝑦 = 𝑌 → (𝜑[𝑌 / 𝑦]𝜑))
5 sbceq1a 3755 . . . . . 6 (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
64, 5sylan9bbr 512 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
7 nfsbc1v 3764 . . . . 5 𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
8 nfcv 2908 . . . . . 6 𝑦𝑋
9 nfsbc1v 3764 . . . . . 6 𝑦[𝑌 / 𝑦]𝜑
108, 9nfsbcw 3766 . . . . 5 𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
111, 2, 3, 6, 7, 10ovmpt3rab1 7616 . . . 4 ((𝑋𝑉𝑌𝑊𝐾𝑈) → (𝑋𝑂𝑌) = (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
1211adantr 482 . . 3 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → (𝑋𝑂𝑌) = (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
1312dmeqd 5866 . 2 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → dom (𝑋𝑂𝑌) = dom (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
14 rabexg 5293 . . . . 5 (𝐿𝑇 → {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
1514adantl 483 . . . 4 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
1615ralrimivw 3148 . . 3 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → ∀𝑧𝐾 {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
17 dmmptg 6199 . . 3 (∀𝑧𝐾 {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V → dom (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}) = 𝐾)
1816, 17syl 17 . 2 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → dom (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}) = 𝐾)
1913, 18eqtrd 2777 1 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → dom (𝑋𝑂𝑌) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3065  {crab 3410  Vcvv 3448  [wsbc 3744  cmpt 5193  dom cdm 5638  (class class class)co 7362  cmpo 7364
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367
This theorem is referenced by:  elovmpt3rab1  7618
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