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 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))

Assertion

Ref	Expression
uspgrsprfv	⊢ (𝑋 ∈ 𝐺 → (𝐹‘𝑋) = (2nd ‘𝑋))

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

Step	Hyp	Ref	Expression
1		uspgrsprf.f	. 2 ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))
2		fveq2 6886	. 2 ⊢ (𝑔 = 𝑋 → (2nd ‘𝑔) = (2nd ‘𝑋))
3		id 22	. 2 ⊢ (𝑋 ∈ 𝐺 → 𝑋 ∈ 𝐺)
4		fvexd 6901	. 2 ⊢ (𝑋 ∈ 𝐺 → (2nd ‘𝑋) ∈ V)
5	1, 2, 3, 4	fvmptd3 7019	1 ⊢ (𝑋 ∈ 𝐺 → (𝐹‘𝑋) = (2nd ‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∃wrex 3059  Vcvv 3463  𝒫 cpw 4580  {copab 5185   ↦ cmpt 5205  ‘cfv 6541  2^nd c2nd 7995  Vtxcvtx 28942  Edgcedg 28993  USPGraphcuspgr 29094  Pairscspr 47437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-iota 6494  df-fun 6543  df-fv 6549

This theorem is referenced by:  uspgrsprf1  48036  uspgrsprfo  48037