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Theorem rfovcnvfvd 44020
Description: Value of the converse of the operator, (𝐴𝑂𝐵), which maps between relations and functions for relations between base sets, 𝐴 and 𝐵, evaluated at function 𝐺. (Contributed by RP, 27-Apr-2021.)
Hypotheses
Ref Expression
rfovd.rf 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦})))
rfovd.a (𝜑𝐴𝑉)
rfovd.b (𝜑𝐵𝑊)
rfovcnvf1od.f 𝐹 = (𝐴𝑂𝐵)
rfovcnvfv.g (𝜑𝐺 ∈ (𝒫 𝐵m 𝐴))
Assertion
Ref Expression
rfovcnvfvd (𝜑 → (𝐹𝐺) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐺𝑥))})
Distinct variable groups:   𝐴,𝑎,𝑏,𝑟,𝑥,𝑦   𝐵,𝑎,𝑏,𝑟,𝑥,𝑦   𝑥,𝐺,𝑦   𝑊,𝑎,𝑥   𝜑,𝑎,𝑏,𝑟,𝑥,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑟,𝑎,𝑏)   𝐺(𝑟,𝑎,𝑏)   𝑂(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑟,𝑎,𝑏)   𝑊(𝑦,𝑟,𝑏)

Proof of Theorem rfovcnvfvd
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 rfovd.rf . . 3 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥𝑎 ↦ {𝑦𝑏𝑥𝑟𝑦})))
2 rfovd.a . . 3 (𝜑𝐴𝑉)
3 rfovd.b . . 3 (𝜑𝐵𝑊)
4 rfovcnvf1od.f . . 3 𝐹 = (𝐴𝑂𝐵)
51, 2, 3, 4rfovcnvd 44018 . 2 (𝜑𝐹 = (𝑔 ∈ (𝒫 𝐵m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑔𝑥))}))
6 fveq1 6905 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑥) = (𝐺𝑥))
76eleq2d 2827 . . . . 5 (𝑔 = 𝐺 → (𝑦 ∈ (𝑔𝑥) ↔ 𝑦 ∈ (𝐺𝑥)))
87anbi2d 630 . . . 4 (𝑔 = 𝐺 → ((𝑥𝐴𝑦 ∈ (𝑔𝑥)) ↔ (𝑥𝐴𝑦 ∈ (𝐺𝑥))))
98opabbidv 5209 . . 3 (𝑔 = 𝐺 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑔𝑥))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐺𝑥))})
109adantl 481 . 2 ((𝜑𝑔 = 𝐺) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑔𝑥))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐺𝑥))})
11 rfovcnvfv.g . 2 (𝜑𝐺 ∈ (𝒫 𝐵m 𝐴))
12 simprl 771 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦 ∈ (𝐺𝑥))) → 𝑥𝐴)
13 elmapi 8889 . . . . . . . 8 (𝐺 ∈ (𝒫 𝐵m 𝐴) → 𝐺:𝐴⟶𝒫 𝐵)
1413ffvelcdmda 7104 . . . . . . 7 ((𝐺 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) → (𝐺𝑥) ∈ 𝒫 𝐵)
1511, 14sylan 580 . . . . . 6 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ 𝒫 𝐵)
1615elpwid 4609 . . . . 5 ((𝜑𝑥𝐴) → (𝐺𝑥) ⊆ 𝐵)
1716sseld 3982 . . . 4 ((𝜑𝑥𝐴) → (𝑦 ∈ (𝐺𝑥) → 𝑦𝐵))
1817impr 454 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦 ∈ (𝐺𝑥))) → 𝑦𝐵)
192, 3, 12, 18opabex2 8082 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐺𝑥))} ∈ V)
205, 10, 11, 19fvmptd 7023 1 (𝜑 → (𝐹𝐺) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐺𝑥))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  𝒫 cpw 4600   class class class wbr 5143  {copab 5205  cmpt 5225   × cxp 5683  ccnv 5684  cfv 6561  (class class class)co 7431  cmpo 7433  m cmap 8866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868
This theorem is referenced by: (None)
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