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Theorem rfovcnvfvd 43997
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 43995 . 2 (𝜑𝐹 = (𝑔 ∈ (𝒫 𝐵m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑔𝑥))}))
6 fveq1 6906 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑥) = (𝐺𝑥))
76eleq2d 2825 . . . . 5 (𝑔 = 𝐺 → (𝑦 ∈ (𝑔𝑥) ↔ 𝑦 ∈ (𝐺𝑥)))
87anbi2d 630 . . . 4 (𝑔 = 𝐺 → ((𝑥𝐴𝑦 ∈ (𝑔𝑥)) ↔ (𝑥𝐴𝑦 ∈ (𝐺𝑥))))
98opabbidv 5214 . . 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 8888 . . . . . . . 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 4614 . . . . 5 ((𝜑𝑥𝐴) → (𝐺𝑥) ⊆ 𝐵)
1716sseld 3994 . . . 4 ((𝜑𝑥𝐴) → (𝑦 ∈ (𝐺𝑥) → 𝑦𝐵))
1817impr 454 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦 ∈ (𝐺𝑥))) → 𝑦𝐵)
192, 3, 12, 18opabex2 8081 . 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 1537  wcel 2106  {crab 3433  Vcvv 3478  𝒫 cpw 4605   class class class wbr 5148  {copab 5210  cmpt 5231   × cxp 5687  ccnv 5688  cfv 6563  (class class class)co 7431  cmpo 7433  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867
This theorem is referenced by: (None)
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