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Theorem uspgrsprfv 48034
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 48040. (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 . 2 𝐹 = (𝑔𝐺 ↦ (2nd𝑔))
2 fveq2 6904 . 2 (𝑔 = 𝑋 → (2nd𝑔) = (2nd𝑋))
3 id 22 . 2 (𝑋𝐺𝑋𝐺)
4 fvexd 6919 . 2 (𝑋𝐺 → (2nd𝑋) ∈ V)
51, 2, 3, 4fvmptd3 7037 1 (𝑋𝐺 → (𝐹𝑋) = (2nd𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wrex 3069  Vcvv 3479  𝒫 cpw 4598  {copab 5203  cmpt 5223  cfv 6559  2nd c2nd 8009  Vtxcvtx 29003  Edgcedg 29054  USPGraphcuspgr 29155  Pairscspr 47437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-iota 6512  df-fun 6561  df-fv 6567
This theorem is referenced by:  uspgrsprf1  48036  uspgrsprfo  48037
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