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Theorem uhgrac 25834

Description: The property of being an undirected hypergraph represented by a class. This representation is useful if the set of vertices and the edge function is/needs not to be known. (Contributed by AV, 1-Jan-2020.)

**Assertion**
Ref	Expression
uhgrac	⊢ (𝐺 ∈ UHGrph → (2^nd ‘𝐺):dom (2^nd ‘𝐺)⟶(𝒫 (1^st ‘𝐺) ∖ {∅}))

Proof of Theorem uhgrac Dummy variables 𝑒 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 4111	. . . . 5 ⊢ (𝑣 = (1^st ‘𝐺) → 𝒫 𝑣 = 𝒫 (1^st ‘𝐺))
2	1	difeq1d 3689	. . . 4 ⊢ (𝑣 = (1^st ‘𝐺) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (1^st ‘𝐺) ∖ {∅}))
3	2	feq3d 5945	. . 3 ⊢ (𝑣 = (1^st ‘𝐺) → (𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅}) ↔ 𝑒:dom 𝑒⟶(𝒫 (1^st ‘𝐺) ∖ {∅})))
4		id 22	. . . 4 ⊢ (𝑒 = (2^nd ‘𝐺) → 𝑒 = (2^nd ‘𝐺))
5		dmeq 5246	. . . 4 ⊢ (𝑒 = (2^nd ‘𝐺) → dom 𝑒 = dom (2^nd ‘𝐺))
6	4, 5	feq12d 5946	. . 3 ⊢ (𝑒 = (2^nd ‘𝐺) → (𝑒:dom 𝑒⟶(𝒫 (1^st ‘𝐺) ∖ {∅}) ↔ (2^nd ‘𝐺):dom (2^nd ‘𝐺)⟶(𝒫 (1^st ‘𝐺) ∖ {∅})))
7	3, 6	elopabi 7120	. 2 ⊢ (𝐺 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})} → (2^nd ‘𝐺):dom (2^nd ‘𝐺)⟶(𝒫 (1^st ‘𝐺) ∖ {∅}))
8		df-uhgra 25821	. 2 ⊢ UHGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}
9	7, 8	eleq2s 2706	1 ⊢ (𝐺 ∈ UHGrph → (2^nd ‘𝐺):dom (2^nd ‘𝐺)⟶(𝒫 (1^st ‘𝐺) ∖ {∅}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∅c0 3874 𝒫 cpw 4108 {csn 4125 {copab 4642 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 1^st c1st 7057 2^nd c2nd 7058 UHGrph cuhg 25819

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-1st 7059 df-2nd 7060 df-uhgra 25821

This theorem is referenced by: (None)

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