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Theorem ovmpt3rab1 6844
Description: The value of an operation defined by the maps-to notation with a function into a class abstraction as a result. The domain of the function and the base set of the class abstraction may depend on the operands, using implicit substitution. (Contributed by AV, 16-Jul-2018.) (Revised 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)
ovmpt3rab1.p ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑𝜓))
ovmpt3rab1.x 𝑥𝜓
ovmpt3rab1.y 𝑦𝜓
Assertion
Ref Expression
ovmpt3rab1 ((𝑋𝑉𝑌𝑊𝐾𝑈) → (𝑋𝑂𝑌) = (𝑧𝐾 ↦ {𝑎𝐿𝜓}))
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
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑎)   𝜓(𝑥,𝑦,𝑧,𝑎)   𝑈(𝑧,𝑎)   𝐾(𝑎)   𝐿(𝑧)   𝑀(𝑥,𝑦,𝑧,𝑎)   𝑁(𝑥,𝑦,𝑧)   𝑂(𝑥,𝑦,𝑧,𝑎)   𝑉(𝑧,𝑎)   𝑊(𝑧,𝑎)

Proof of Theorem ovmpt3rab1
StepHypRef Expression
1 ovmpt3rab1.o . . 3 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧𝑀 ↦ {𝑎𝑁𝜑}))
21a1i 11 . 2 ((𝑋𝑉𝑌𝑊𝐾𝑈) → 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧𝑀 ↦ {𝑎𝑁𝜑})))
3 ovmpt3rab1.m . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑀 = 𝐾)
4 ovmpt3rab1.n . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑁 = 𝐿)
5 ovmpt3rab1.p . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑𝜓))
64, 5rabeqbidv 3181 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → {𝑎𝑁𝜑} = {𝑎𝐿𝜓})
73, 6mpteq12dv 4693 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑧𝑀 ↦ {𝑎𝑁𝜑}) = (𝑧𝐾 ↦ {𝑎𝐿𝜓}))
87adantl 482 . 2 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑧𝑀 ↦ {𝑎𝑁𝜑}) = (𝑧𝐾 ↦ {𝑎𝐿𝜓}))
9 eqidd 2622 . 2 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝑥 = 𝑋) → V = V)
10 elex 3198 . . 3 (𝑋𝑉𝑋 ∈ V)
11103ad2ant1 1080 . 2 ((𝑋𝑉𝑌𝑊𝐾𝑈) → 𝑋 ∈ V)
12 elex 3198 . . 3 (𝑌𝑊𝑌 ∈ V)
13123ad2ant2 1081 . 2 ((𝑋𝑉𝑌𝑊𝐾𝑈) → 𝑌 ∈ V)
14 mptexg 6438 . . 3 (𝐾𝑈 → (𝑧𝐾 ↦ {𝑎𝐿𝜓}) ∈ V)
15143ad2ant3 1082 . 2 ((𝑋𝑉𝑌𝑊𝐾𝑈) → (𝑧𝐾 ↦ {𝑎𝐿𝜓}) ∈ V)
16 nfv 1840 . 2 𝑥(𝑋𝑉𝑌𝑊𝐾𝑈)
17 nfv 1840 . 2 𝑦(𝑋𝑉𝑌𝑊𝐾𝑈)
18 nfcv 2761 . 2 𝑦𝑋
19 nfcv 2761 . 2 𝑥𝑌
20 nfcv 2761 . . 3 𝑥𝐾
21 ovmpt3rab1.x . . . 4 𝑥𝜓
22 nfcv 2761 . . . 4 𝑥𝐿
2321, 22nfrab 3112 . . 3 𝑥{𝑎𝐿𝜓}
2420, 23nfmpt 4706 . 2 𝑥(𝑧𝐾 ↦ {𝑎𝐿𝜓})
25 nfcv 2761 . . 3 𝑦𝐾
26 ovmpt3rab1.y . . . 4 𝑦𝜓
27 nfcv 2761 . . . 4 𝑦𝐿
2826, 27nfrab 3112 . . 3 𝑦{𝑎𝐿𝜓}
2925, 28nfmpt 4706 . 2 𝑦(𝑧𝐾 ↦ {𝑎𝐿𝜓})
302, 8, 9, 11, 13, 15, 16, 17, 18, 19, 24, 29ovmpt2dxf 6739 1 ((𝑋𝑉𝑌𝑊𝐾𝑈) → (𝑋𝑂𝑌) = (𝑧𝐾 ↦ {𝑎𝐿𝜓}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wnf 1705  wcel 1987  {crab 2911  Vcvv 3186  cmpt 4673  (class class class)co 6604  cmpt2 6606
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 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867
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 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609
This theorem is referenced by:  ovmpt3rabdm  6845  elovmpt3rab1  6846
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