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Theorem ovmpt3rabdm 7674
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 3781 . . . . . 6 (𝑦 = 𝑌 → (𝜑[𝑌 / 𝑦]𝜑))
5 sbceq1a 3781 . . . . . 6 (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
64, 5sylan9bbr 509 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
7 nfsbc1v 3790 . . . . 5 𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
8 nfcv 2892 . . . . . 6 𝑦𝑋
9 nfsbc1v 3790 . . . . . 6 𝑦[𝑌 / 𝑦]𝜑
108, 9nfsbcw 3792 . . . . 5 𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
111, 2, 3, 6, 7, 10ovmpt3rab1 7673 . . . 4 ((𝑋𝑉𝑌𝑊𝐾𝑈) → (𝑋𝑂𝑌) = (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
1211adantr 479 . . 3 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → (𝑋𝑂𝑌) = (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
1312dmeqd 5903 . 2 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → dom (𝑋𝑂𝑌) = dom (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
14 rabexg 5329 . . . . 5 (𝐿𝑇 → {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
1514adantl 480 . . . 4 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
1615ralrimivw 3140 . . 3 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → ∀𝑧𝐾 {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
17 dmmptg 6242 . . 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 2765 1 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → dom (𝑋𝑂𝑌) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3051  {crab 3419  Vcvv 3463  [wsbc 3770  cmpt 5227  dom cdm 5673  (class class class)co 7413  cmpo 7415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7416  df-oprab 7417  df-mpo 7418
This theorem is referenced by:  elovmpt3rab1  7675
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