uspgrsprfv - Mathbox for Alexander van der Vekens

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Theorem uspgrsprfv 47207

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 47213. (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 ↦ (2^nd ‘𝑔))

**Assertion**
Ref	Expression
uspgrsprfv	⊢ (𝑋 ∈ 𝐺 → (𝐹‘𝑋) = (2^nd ‘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 ↦ (2^nd ‘𝑔))
2		fveq2 6897	. 2 ⊢ (𝑔 = 𝑋 → (2^nd ‘𝑔) = (2^nd ‘𝑋))
3		id 22	. 2 ⊢ (𝑋 ∈ 𝐺 → 𝑋 ∈ 𝐺)
4		fvexd 6912	. 2 ⊢ (𝑋 ∈ 𝐺 → (2^nd ‘𝑋) ∈ V)
5	1, 2, 3, 4	fvmptd3 7028	1 ⊢ (𝑋 ∈ 𝐺 → (𝐹‘𝑋) = (2^nd ‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃wrex 3067 Vcvv 3471 𝒫 cpw 4603 {copab 5210 ↦ cmpt 5231 ‘cfv 6548 2^nd c2nd 7992 Vtxcvtx 28822 Edgcedg 28873 USPGraphcuspgr 28974 Pairscspr 46817

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556

This theorem is referenced by: uspgrsprf1 47209 uspgrsprfo 47210

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