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Theorem rfovfvd 42348
Description: Value of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵, and relation 𝑅. (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)
rfovfvd.r (𝜑𝑅 ∈ 𝒫 (𝐴 × 𝐵))
rfovfvd.f 𝐹 = (𝐴𝑂𝐵)
Assertion
Ref Expression
rfovfvd (𝜑 → (𝐹𝑅) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑅𝑦}))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑟,𝑥   𝐵,𝑎,𝑏,𝑟,𝑥   𝑦,𝐵,𝑎,𝑏,𝑟   𝑅,𝑟,𝑥   𝑦,𝑅   𝜑,𝑎,𝑏,𝑟
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)   𝑅(𝑎,𝑏)   𝐹(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑂(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑟,𝑎,𝑏)

Proof of Theorem rfovfvd
StepHypRef Expression
1 rfovfvd.f . . 3 𝐹 = (𝐴𝑂𝐵)
2 rfovd.rf . . . 4 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦})))
3 rfovd.a . . . 4 (𝜑𝐴𝑉)
4 rfovd.b . . . 4 (𝜑𝐵𝑊)
52, 3, 4rfovd 42347 . . 3 (𝜑 → (𝐴𝑂𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
61, 5eqtrid 2789 . 2 (𝜑𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
7 breq 5112 . . . . 5 (𝑟 = 𝑅 → (𝑥𝑟𝑦𝑥𝑅𝑦))
87rabbidv 3418 . . . 4 (𝑟 = 𝑅 → {𝑦𝐵𝑥𝑟𝑦} = {𝑦𝐵𝑥𝑅𝑦})
98mpteq2dv 5212 . . 3 (𝑟 = 𝑅 → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑅𝑦}))
109adantl 483 . 2 ((𝜑𝑟 = 𝑅) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑅𝑦}))
11 rfovfvd.r . 2 (𝜑𝑅 ∈ 𝒫 (𝐴 × 𝐵))
123mptexd 7179 . 2 (𝜑 → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑅𝑦}) ∈ V)
136, 10, 11, 12fvmptd 6960 1 (𝜑 → (𝐹𝑅) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑅𝑦}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {crab 3410  Vcvv 3448  𝒫 cpw 4565   class class class wbr 5110  cmpt 5193   × cxp 5636  cfv 6501  (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-pow 5325  ax-pr 5389  ax-un 7677
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-pw 4567  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:  rfovfvfvd  42349
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