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Theorem rfovcnvfvd 38803
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 38801 . 2 (𝜑𝐹 = (𝑔 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑔𝑥))}))
6 fveq1 6351 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑥) = (𝐺𝑥))
76eleq2d 2825 . . . . 5 (𝑔 = 𝐺 → (𝑦 ∈ (𝑔𝑥) ↔ 𝑦 ∈ (𝐺𝑥)))
87anbi2d 742 . . . 4 (𝑔 = 𝐺 → ((𝑥𝐴𝑦 ∈ (𝑔𝑥)) ↔ (𝑥𝐴𝑦 ∈ (𝐺𝑥))))
98opabbidv 4868 . . 3 (𝑔 = 𝐺 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑔𝑥))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐺𝑥))})
109adantl 473 . 2 ((𝜑𝑔 = 𝐺) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑔𝑥))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐺𝑥))})
11 rfovcnvfv.g . 2 (𝜑𝐺 ∈ (𝒫 𝐵𝑚 𝐴))
12 simprl 811 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦 ∈ (𝐺𝑥))) → 𝑥𝐴)
13 elmapi 8045 . . . . . . . 8 (𝐺 ∈ (𝒫 𝐵𝑚 𝐴) → 𝐺:𝐴⟶𝒫 𝐵)
1413ffvelrnda 6522 . . . . . . 7 ((𝐺 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑥𝐴) → (𝐺𝑥) ∈ 𝒫 𝐵)
1511, 14sylan 489 . . . . . 6 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ 𝒫 𝐵)
1615elpwid 4314 . . . . 5 ((𝜑𝑥𝐴) → (𝐺𝑥) ⊆ 𝐵)
1716sseld 3743 . . . 4 ((𝜑𝑥𝐴) → (𝑦 ∈ (𝐺𝑥) → 𝑦𝐵))
1817impr 650 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦 ∈ (𝐺𝑥))) → 𝑦𝐵)
192, 3, 12, 18opabex2 7394 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐺𝑥))} ∈ V)
205, 10, 11, 19fvmptd 6450 1 (𝜑 → (𝐹𝐺) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐺𝑥))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  𝒫 cpw 4302   class class class wbr 4804  {copab 4864  cmpt 4881   × cxp 5264  ccnv 5265  cfv 6049  (class class class)co 6813  cmpt2 6815  𝑚 cmap 8023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025
This theorem is referenced by: (None)
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