Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovmpt3rab1 Structured version   Visualization version   GIF version

Theorem ovmpt3rab1 6789
 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 3168 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → {𝑎𝑁𝜑} = {𝑎𝐿𝜓})
73, 6mpteq12dv 4663 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑧𝑀 ↦ {𝑎𝑁𝜑}) = (𝑧𝐾 ↦ {𝑎𝐿𝜓}))
87adantl 481 . 2 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑧𝑀 ↦ {𝑎𝑁𝜑}) = (𝑧𝐾 ↦ {𝑎𝐿𝜓}))
9 eqidd 2611 . 2 (((𝑋𝑉𝑌𝑊𝐾𝑈) ∧ 𝑥 = 𝑋) → V = V)
10 elex 3185 . . 3 (𝑋𝑉𝑋 ∈ V)
11103ad2ant1 1075 . 2 ((𝑋𝑉𝑌𝑊𝐾𝑈) → 𝑋 ∈ V)
12 elex 3185 . . 3 (𝑌𝑊𝑌 ∈ V)
13123ad2ant2 1076 . 2 ((𝑋𝑉𝑌𝑊𝐾𝑈) → 𝑌 ∈ V)
14 mptexg 6389 . . 3 (𝐾𝑈 → (𝑧𝐾 ↦ {𝑎𝐿𝜓}) ∈ V)
15143ad2ant3 1077 . 2 ((𝑋𝑉𝑌𝑊𝐾𝑈) → (𝑧𝐾 ↦ {𝑎𝐿𝜓}) ∈ V)
16 nfv 1830 . 2 𝑥(𝑋𝑉𝑌𝑊𝐾𝑈)
17 nfv 1830 . 2 𝑦(𝑋𝑉𝑌𝑊𝐾𝑈)
18 nfcv 2751 . 2 𝑦𝑋
19 nfcv 2751 . 2 𝑥𝑌
20 nfcv 2751 . . 3 𝑥𝐾
21 ovmpt3rab1.x . . . 4 𝑥𝜓
22 nfcv 2751 . . . 4 𝑥𝐿
2321, 22nfrab 3100 . . 3 𝑥{𝑎𝐿𝜓}
2420, 23nfmpt 4674 . 2 𝑥(𝑧𝐾 ↦ {𝑎𝐿𝜓})
25 nfcv 2751 . . 3 𝑦𝐾
26 ovmpt3rab1.y . . . 4 𝑦𝜓
27 nfcv 2751 . . . 4 𝑦𝐿
2826, 27nfrab 3100 . . 3 𝑦{𝑎𝐿𝜓}
2925, 28nfmpt 4674 . 2 𝑦(𝑧𝐾 ↦ {𝑎𝐿𝜓})
302, 8, 9, 11, 13, 15, 16, 17, 18, 19, 24, 29ovmpt2dxf 6684 1 ((𝑋𝑉𝑌𝑊𝐾𝑈) → (𝑋𝑂𝑌) = (𝑧𝐾 ↦ {𝑎𝐿𝜓}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  {crab 2900  Vcvv 3173   ↦ cmpt 4643  (class class class)co 6549   ↦ cmpt2 6551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  ovmpt3rabdm  6790  elovmpt3rab1  6791
 Copyright terms: Public domain W3C validator