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Theorem ovmpt3rab1 7616
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 3427 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → {𝑎𝑁𝜑} = {𝑎𝐿𝜓})
73, 6mpteq12dv 5201 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑧𝑀 ↦ {𝑎𝑁𝜑}) = (𝑧𝐾 ↦ {𝑎𝐿𝜓}))
87adantl 483 . 2 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑧𝑀 ↦ {𝑎𝑁𝜑}) = (𝑧𝐾 ↦ {𝑎𝐿𝜓}))
9 eqidd 2738 . 2 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝑥 = 𝑋) → V = V)
10 elex 3466 . . 3 (𝑋𝑉𝑋 ∈ V)
11103ad2ant1 1134 . 2 ((𝑋𝑉𝑌𝑊𝐾𝑈) → 𝑋 ∈ V)
12 elex 3466 . . 3 (𝑌𝑊𝑌 ∈ V)
13123ad2ant2 1135 . 2 ((𝑋𝑉𝑌𝑊𝐾𝑈) → 𝑌 ∈ V)
14 mptexg 7176 . . 3 (𝐾𝑈 → (𝑧𝐾 ↦ {𝑎𝐿𝜓}) ∈ V)
15143ad2ant3 1136 . 2 ((𝑋𝑉𝑌𝑊𝐾𝑈) → (𝑧𝐾 ↦ {𝑎𝐿𝜓}) ∈ V)
16 nfv 1918 . 2 𝑥(𝑋𝑉𝑌𝑊𝐾𝑈)
17 nfv 1918 . 2 𝑦(𝑋𝑉𝑌𝑊𝐾𝑈)
18 nfcv 2908 . 2 𝑦𝑋
19 nfcv 2908 . 2 𝑥𝑌
20 nfcv 2908 . . 3 𝑥𝐾
21 ovmpt3rab1.x . . . 4 𝑥𝜓
22 nfcv 2908 . . . 4 𝑥𝐿
2321, 22nfrabw 3443 . . 3 𝑥{𝑎𝐿𝜓}
2420, 23nfmpt 5217 . 2 𝑥(𝑧𝐾 ↦ {𝑎𝐿𝜓})
25 nfcv 2908 . . 3 𝑦𝐾
26 ovmpt3rab1.y . . . 4 𝑦𝜓
27 nfcv 2908 . . . 4 𝑦𝐿
2826, 27nfrabw 3443 . . 3 𝑦{𝑎𝐿𝜓}
2925, 28nfmpt 5217 . 2 𝑦(𝑧𝐾 ↦ {𝑎𝐿𝜓})
302, 8, 9, 11, 13, 15, 16, 17, 18, 19, 24, 29ovmpodxf 7510 1 ((𝑋𝑉𝑌𝑊𝐾𝑈) → (𝑋𝑂𝑌) = (𝑧𝐾 ↦ {𝑎𝐿𝜓}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wnf 1786  wcel 2107  {crab 3410  Vcvv 3448  cmpt 5193  (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:  ovmpt3rabdm  7617  elovmpt3rab1  7618
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