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Theorem rfovcnvfvd 42744
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 42742 . 2 (𝜑𝐹 = (𝑔 ∈ (𝒫 𝐵m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑔𝑥))}))
6 fveq1 6888 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑥) = (𝐺𝑥))
76eleq2d 2820 . . . . 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 483 . 2 ((𝜑𝑔 = 𝐺) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑔𝑥))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐺𝑥))})
11 rfovcnvfv.g . 2 (𝜑𝐺 ∈ (𝒫 𝐵m 𝐴))
12 simprl 770 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦 ∈ (𝐺𝑥))) → 𝑥𝐴)
13 elmapi 8840 . . . . . . . 8 (𝐺 ∈ (𝒫 𝐵m 𝐴) → 𝐺:𝐴⟶𝒫 𝐵)
1413ffvelcdmda 7084 . . . . . . 7 ((𝐺 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) → (𝐺𝑥) ∈ 𝒫 𝐵)
1511, 14sylan 581 . . . . . 6 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ 𝒫 𝐵)
1615elpwid 4611 . . . . 5 ((𝜑𝑥𝐴) → (𝐺𝑥) ⊆ 𝐵)
1716sseld 3981 . . . 4 ((𝜑𝑥𝐴) → (𝑦 ∈ (𝐺𝑥) → 𝑦𝐵))
1817impr 456 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦 ∈ (𝐺𝑥))) → 𝑦𝐵)
192, 3, 12, 18opabex2 8040 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐺𝑥))} ∈ V)
205, 10, 11, 19fvmptd 7003 1 (𝜑 → (𝐹𝐺) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐺𝑥))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {crab 3433  Vcvv 3475  𝒫 cpw 4602   class class class wbr 5148  {copab 5210  cmpt 5231   × cxp 5674  ccnv 5675  cfv 6541  (class class class)co 7406  cmpo 7408  m cmap 8817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1st 7972  df-2nd 7973  df-map 8819
This theorem is referenced by: (None)
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