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Theorem rfovd 43302
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 5692 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎 × 𝑏) = (𝐴 × 𝐵))
43pweqd 4612 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝒫 (𝑎 × 𝑏) = 𝒫 (𝐴 × 𝐵))
5 simpl 482 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝑎 = 𝐴)
6 rabeq 3438 . . . . . 6 (𝑏 = 𝐵 → {𝑦𝑏𝑥𝑟𝑦} = {𝑦𝐵𝑥𝑟𝑦})
76adantl 481 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → {𝑦𝑏𝑥𝑟𝑦} = {𝑦𝐵𝑥𝑟𝑦})
85, 7mpteq12dv 5230 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦}) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
94, 8mpteq12dv 5230 . . 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 3487 . 2 (𝜑𝐴 ∈ V)
13 rfovd.b . . 3 (𝜑𝐵𝑊)
1413elexd 3487 . 2 (𝜑𝐵 ∈ V)
1511, 13xpexd 7732 . . 3 (𝜑 → (𝐴 × 𝐵) ∈ V)
16 pwexg 5367 . . 3 ((𝐴 × 𝐵) ∈ V → 𝒫 (𝐴 × 𝐵) ∈ V)
17 mptexg 7215 . . 3 (𝒫 (𝐴 × 𝐵) ∈ V → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∈ V)
1815, 16, 173syl 18 . 2 (𝜑 → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∈ V)
192, 10, 12, 14, 18ovmpod 7553 1 (𝜑 → (𝐴𝑂𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  {crab 3424  Vcvv 3466  𝒫 cpw 4595   class class class wbr 5139  cmpt 5222   × cxp 5665  (class class class)co 7402  cmpo 7404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407
This theorem is referenced by:  rfovfvd  43303  rfovcnvf1od  43305  fsovrfovd  43310
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