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Theorem rfovcnvfvd 37780
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 37778 . 2 (𝜑𝐹 = (𝑔 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑔𝑥))}))
6 fveq1 6147 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑥) = (𝐺𝑥))
76eleq2d 2684 . . . . 5 (𝑔 = 𝐺 → (𝑦 ∈ (𝑔𝑥) ↔ 𝑦 ∈ (𝐺𝑥)))
87anbi2d 739 . . . 4 (𝑔 = 𝐺 → ((𝑥𝐴𝑦 ∈ (𝑔𝑥)) ↔ (𝑥𝐴𝑦 ∈ (𝐺𝑥))))
98opabbidv 4678 . . 3 (𝑔 = 𝐺 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑔𝑥))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐺𝑥))})
109adantl 482 . 2 ((𝜑𝑔 = 𝐺) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑔𝑥))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐺𝑥))})
11 rfovcnvfv.g . 2 (𝜑𝐺 ∈ (𝒫 𝐵𝑚 𝐴))
12 simprl 793 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦 ∈ (𝐺𝑥))) → 𝑥𝐴)
13 elmapi 7823 . . . . . . . 8 (𝐺 ∈ (𝒫 𝐵𝑚 𝐴) → 𝐺:𝐴⟶𝒫 𝐵)
1413ffvelrnda 6315 . . . . . . 7 ((𝐺 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑥𝐴) → (𝐺𝑥) ∈ 𝒫 𝐵)
1511, 14sylan 488 . . . . . 6 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ 𝒫 𝐵)
1615elpwid 4141 . . . . 5 ((𝜑𝑥𝐴) → (𝐺𝑥) ⊆ 𝐵)
1716sseld 3582 . . . 4 ((𝜑𝑥𝐴) → (𝑦 ∈ (𝐺𝑥) → 𝑦𝐵))
1817impr 648 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦 ∈ (𝐺𝑥))) → 𝑦𝐵)
192, 3, 12, 18opabex2 7172 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐺𝑥))} ∈ V)
205, 10, 11, 19fvmptd 6245 1 (𝜑 → (𝐹𝐺) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐺𝑥))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186  𝒫 cpw 4130   class class class wbr 4613  {copab 4672  cmpt 4673   × cxp 5072  ccnv 5073  cfv 5847  (class class class)co 6604  cmpt2 6606  𝑚 cmap 7802
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804
This theorem is referenced by: (None)
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