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Theorem uspgrsprfv 42552
 Description: The value of the function 𝐹 which maps a "simple pseudograph" for a fixed set 𝑉 of vertices to the set of edges (i.e. range of the edge function) of the graph. Solely for 𝐺 as defined here, the function 𝐹 is a bijection between the "simple pseudographs" and the subsets of the set of pairs 𝑃 over the fixed set 𝑉 of vertices, see uspgrbispr 42558. (Contributed by AV, 24-Nov-2021.)
Hypotheses
Ref Expression
uspgrsprf.p 𝑃 = 𝒫 (Pairs‘𝑉)
uspgrsprf.g 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}
uspgrsprf.f 𝐹 = (𝑔𝐺 ↦ (2nd𝑔))
Assertion
Ref Expression
uspgrsprfv (𝑋𝐺 → (𝐹𝑋) = (2nd𝑋))
Distinct variable groups:   𝑃,𝑒,𝑞,𝑣   𝑒,𝑉,𝑞,𝑣   𝑔,𝐺   𝑔,𝑋
Allowed substitution hints:   𝑃(𝑔)   𝐹(𝑣,𝑒,𝑔,𝑞)   𝐺(𝑣,𝑒,𝑞)   𝑉(𝑔)   𝑋(𝑣,𝑒,𝑞)

Proof of Theorem uspgrsprfv
StepHypRef Expression
1 uspgrsprf.f . . 3 𝐹 = (𝑔𝐺 ↦ (2nd𝑔))
21a1i 11 . 2 (𝑋𝐺𝐹 = (𝑔𝐺 ↦ (2nd𝑔)))
3 fveq2 6411 . . 3 (𝑔 = 𝑋 → (2nd𝑔) = (2nd𝑋))
43adantl 474 . 2 ((𝑋𝐺𝑔 = 𝑋) → (2nd𝑔) = (2nd𝑋))
5 id 22 . 2 (𝑋𝐺𝑋𝐺)
6 fvexd 6426 . 2 (𝑋𝐺 → (2nd𝑋) ∈ V)
72, 4, 5, 6fvmptd 6513 1 (𝑋𝐺 → (𝐹𝑋) = (2nd𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157  ∃wrex 3090  Vcvv 3385  𝒫 cpw 4349  {copab 4905   ↦ cmpt 4922  ‘cfv 6101  2nd c2nd 7400  Vtxcvtx 26231  Edgcedg 26282  USPGraphcuspgr 26384  Pairscspr 42526 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109 This theorem is referenced by:  uspgrsprf1  42554  uspgrsprfo  42555
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