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Theorem rfovd 37315
 Description: Value of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵. (Contributed by RP, 25-Apr-2021.)
Hypotheses
Ref Expression
rfovd.rf 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦})))
rfovd.a (𝜑𝐴𝑉)
rfovd.b (𝜑𝐵𝑊)
Assertion
Ref Expression
rfovd (𝜑 → (𝐴𝑂𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑟   𝑥,𝐴,𝑎,𝑏   𝐵,𝑎,𝑏,𝑟   𝑥,𝐵   𝑦,𝐵,𝑎,𝑏   𝜑,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑟)   𝐴(𝑦)   𝑂(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑟,𝑎,𝑏)

Proof of Theorem rfovd
StepHypRef Expression
1 rfovd.rf . . 3 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦})))
21a1i 11 . 2 (𝜑𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦}))))
3 xpeq12 5058 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎 × 𝑏) = (𝐴 × 𝐵))
43pweqd 4113 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝒫 (𝑎 × 𝑏) = 𝒫 (𝐴 × 𝐵))
5 simpl 472 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝑎 = 𝐴)
6 rabeq 3166 . . . . . 6 (𝑏 = 𝐵 → {𝑦𝑏𝑥𝑟𝑦} = {𝑦𝐵𝑥𝑟𝑦})
76adantl 481 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → {𝑦𝑏𝑥𝑟𝑦} = {𝑦𝐵𝑥𝑟𝑦})
85, 7mpteq12dv 4663 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦}) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
94, 8mpteq12dv 4663 . . 3 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦})) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
109adantl 481 . 2 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦})) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
11 rfovd.a . . 3 (𝜑𝐴𝑉)
1211elexd 3187 . 2 (𝜑𝐴 ∈ V)
13 rfovd.b . . 3 (𝜑𝐵𝑊)
1413elexd 3187 . 2 (𝜑𝐵 ∈ V)
15 xpexg 6858 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
1611, 13, 15syl2anc 691 . . 3 (𝜑 → (𝐴 × 𝐵) ∈ V)
17 pwexg 4776 . . 3 ((𝐴 × 𝐵) ∈ V → 𝒫 (𝐴 × 𝐵) ∈ V)
18 mptexg 6389 . . 3 (𝒫 (𝐴 × 𝐵) ∈ V → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∈ V)
1916, 17, 183syl 18 . 2 (𝜑 → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∈ V)
202, 10, 12, 14, 19ovmpt2d 6686 1 (𝜑 → (𝐴𝑂𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  𝒫 cpw 4108   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  (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-8 1979  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-pow 4769  ax-pr 4833  ax-un 6847 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-pw 4110  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:  rfovfvd  37316  rfovcnvf1od  37318  fsovrfovd  37323
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