Theorem uspgrsprfv 48255

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 48261. (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 6822	. 2 ⊢ (𝑔 = 𝑋 → (2nd ‘𝑔) = (2nd ‘𝑋))
3		id 22	. 2 ⊢ (𝑋 ∈ 𝐺 → 𝑋 ∈ 𝐺)
4		fvexd 6837	. 2 ⊢ (𝑋 ∈ 𝐺 → (2nd ‘𝑋) ∈ V)
5	1, 2, 3, 4	fvmptd3 6952	1 ⊢ (𝑋 ∈ 𝐺 → (𝐹‘𝑋) = (2nd ‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  Vcvv 3436  𝒫 cpw 4547  {copab 5151   ↦ cmpt 5170  ‘cfv 6481  2^nd c2nd 7920  Vtxcvtx 28974  Edgcedg 29025  USPGraphcuspgr 29126  Pairscspr 47587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fv 6489

This theorem is referenced by:  uspgrsprf1  48257  uspgrsprfo  48258