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Theorem ovmpt3rabdm 6852
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 3432 . . . . . 6 (𝑦 = 𝑌 → (𝜑[𝑌 / 𝑦]𝜑))
5 sbceq1a 3432 . . . . . 6 (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
64, 5sylan9bbr 736 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
7 nfsbc1v 3441 . . . . 5 𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
8 nfcv 2761 . . . . . 6 𝑦𝑋
9 nfsbc1v 3441 . . . . . 6 𝑦[𝑌 / 𝑦]𝜑
108, 9nfsbc 3443 . . . . 5 𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
111, 2, 3, 6, 7, 10ovmpt3rab1 6851 . . . 4 ((𝑋𝑉𝑌𝑊𝐾𝑈) → (𝑋𝑂𝑌) = (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
1211adantr 481 . . 3 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → (𝑋𝑂𝑌) = (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
1312dmeqd 5291 . 2 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → dom (𝑋𝑂𝑌) = dom (𝑧𝐾 ↦ {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
14 rabexg 4777 . . . . 5 (𝐿𝑇 → {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
1514adantl 482 . . . 4 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
1615ralrimivw 2962 . . 3 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → ∀𝑧𝐾 {𝑎𝐿[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
17 dmmptg 5596 . . 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 2655 1 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝐿𝑇) → dom (𝑋𝑂𝑌) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3189  [wsbc 3421  cmpt 4678  dom cdm 5079  (class class class)co 6610  cmpt2 6612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615
This theorem is referenced by:  elovmpt3rab1  6853
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