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Theorem ovmpt3rab1 7388
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 3461 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → {𝑎𝑁𝜑} = {𝑎𝐿𝜓})
73, 6mpteq12dv 5127 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑧𝑀 ↦ {𝑎𝑁𝜑}) = (𝑧𝐾 ↦ {𝑎𝐿𝜓}))
87adantl 485 . 2 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑧𝑀 ↦ {𝑎𝑁𝜑}) = (𝑧𝐾 ↦ {𝑎𝐿𝜓}))
9 eqidd 2823 . 2 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝑥 = 𝑋) → V = V)
10 elex 3487 . . 3 (𝑋𝑉𝑋 ∈ V)
11103ad2ant1 1130 . 2 ((𝑋𝑉𝑌𝑊𝐾𝑈) → 𝑋 ∈ V)
12 elex 3487 . . 3 (𝑌𝑊𝑌 ∈ V)
13123ad2ant2 1131 . 2 ((𝑋𝑉𝑌𝑊𝐾𝑈) → 𝑌 ∈ V)
14 mptexg 6966 . . 3 (𝐾𝑈 → (𝑧𝐾 ↦ {𝑎𝐿𝜓}) ∈ V)
15143ad2ant3 1132 . 2 ((𝑋𝑉𝑌𝑊𝐾𝑈) → (𝑧𝐾 ↦ {𝑎𝐿𝜓}) ∈ V)
16 nfv 1915 . 2 𝑥(𝑋𝑉𝑌𝑊𝐾𝑈)
17 nfv 1915 . 2 𝑦(𝑋𝑉𝑌𝑊𝐾𝑈)
18 nfcv 2979 . 2 𝑦𝑋
19 nfcv 2979 . 2 𝑥𝑌
20 nfcv 2979 . . 3 𝑥𝐾
21 ovmpt3rab1.x . . . 4 𝑥𝜓
22 nfcv 2979 . . . 4 𝑥𝐿
2321, 22nfrabw 3366 . . 3 𝑥{𝑎𝐿𝜓}
2420, 23nfmpt 5139 . 2 𝑥(𝑧𝐾 ↦ {𝑎𝐿𝜓})
25 nfcv 2979 . . 3 𝑦𝐾
26 ovmpt3rab1.y . . . 4 𝑦𝜓
27 nfcv 2979 . . . 4 𝑦𝐿
2826, 27nfrabw 3366 . . 3 𝑦{𝑎𝐿𝜓}
2925, 28nfmpt 5139 . 2 𝑦(𝑧𝐾 ↦ {𝑎𝐿𝜓})
302, 8, 9, 11, 13, 15, 16, 17, 18, 19, 24, 29ovmpodxf 7284 1 ((𝑋𝑉𝑌𝑊𝐾𝑈) → (𝑋𝑂𝑌) = (𝑧𝐾 ↦ {𝑎𝐿𝜓}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wnf 1785  wcel 2114  {crab 3134  Vcvv 3469  cmpt 5122  (class class class)co 7140  cmpo 7142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145
This theorem is referenced by:  ovmpt3rabdm  7389  elovmpt3rab1  7390
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