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Theorem rfovfvfvd 43498
Description: Value of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵, relation 𝑅, and left element 𝑋. (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)
rfovfvfvd.x (𝜑𝑋𝐴)
rfovfvfvd.g 𝐺 = (𝐹𝑅)
Assertion
Ref Expression
rfovfvfvd (𝜑 → (𝐺𝑋) = {𝑦𝐵𝑋𝑅𝑦})
Distinct variable groups:   𝐴,𝑎,𝑏,𝑟,𝑥   𝐵,𝑎,𝑏,𝑟,𝑥,𝑦   𝑅,𝑟,𝑥,𝑦   𝑥,𝑋,𝑦   𝜑,𝑎,𝑏,𝑟,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝑅(𝑎,𝑏)   𝐹(𝑥,𝑦,𝑟,𝑎,𝑏)   𝐺(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑂(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑋(𝑟,𝑎,𝑏)

Proof of Theorem rfovfvfvd
StepHypRef Expression
1 rfovfvfvd.g . . 3 𝐺 = (𝐹𝑅)
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)
5 rfovfvd.r . . . 4 (𝜑𝑅 ∈ 𝒫 (𝐴 × 𝐵))
6 rfovfvd.f . . . 4 𝐹 = (𝐴𝑂𝐵)
72, 3, 4, 5, 6rfovfvd 43497 . . 3 (𝜑 → (𝐹𝑅) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑅𝑦}))
81, 7eqtrid 2777 . 2 (𝜑𝐺 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑅𝑦}))
9 breq1 5146 . . . 4 (𝑥 = 𝑋 → (𝑥𝑅𝑦𝑋𝑅𝑦))
109rabbidv 3427 . . 3 (𝑥 = 𝑋 → {𝑦𝐵𝑥𝑅𝑦} = {𝑦𝐵𝑋𝑅𝑦})
1110adantl 480 . 2 ((𝜑𝑥 = 𝑋) → {𝑦𝐵𝑥𝑅𝑦} = {𝑦𝐵𝑋𝑅𝑦})
12 rfovfvfvd.x . 2 (𝜑𝑋𝐴)
13 rabexg 5328 . . 3 (𝐵𝑊 → {𝑦𝐵𝑋𝑅𝑦} ∈ V)
144, 13syl 17 . 2 (𝜑 → {𝑦𝐵𝑋𝑅𝑦} ∈ V)
158, 11, 12, 14fvmptd 7007 1 (𝜑 → (𝐺𝑋) = {𝑦𝐵𝑋𝑅𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {crab 3419  Vcvv 3463  𝒫 cpw 4598   class class class wbr 5143  cmpt 5226   × cxp 5670  cfv 6543  (class class class)co 7416  cmpo 7418
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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421
This theorem is referenced by: (None)
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