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Theorem rfovfvfvd 37814
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 37813 . . 3 (𝜑 → (𝐹𝑅) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑅𝑦}))
81, 7syl5eq 2667 . 2 (𝜑𝐺 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑅𝑦}))
9 breq1 4621 . . . 4 (𝑥 = 𝑋 → (𝑥𝑅𝑦𝑋𝑅𝑦))
109rabbidv 3180 . . 3 (𝑥 = 𝑋 → {𝑦𝐵𝑥𝑅𝑦} = {𝑦𝐵𝑋𝑅𝑦})
1110adantl 482 . 2 ((𝜑𝑥 = 𝑋) → {𝑦𝐵𝑥𝑅𝑦} = {𝑦𝐵𝑋𝑅𝑦})
12 rfovfvfvd.x . 2 (𝜑𝑋𝐴)
13 rabexg 4777 . . 3 (𝐵𝑊 → {𝑦𝐵𝑋𝑅𝑦} ∈ V)
144, 13syl 17 . 2 (𝜑 → {𝑦𝐵𝑋𝑅𝑦} ∈ V)
158, 11, 12, 14fvmptd 6250 1 (𝜑 → (𝐺𝑋) = {𝑦𝐵𝑋𝑅𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3189  𝒫 cpw 4135   class class class wbr 4618  cmpt 4678   × cxp 5077  cfv 5852  (class class class)co 6610  cmpt2 6612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615
This theorem is referenced by: (None)
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