Theorem 1loopgredg 29760

Description: The set of edges in a graph (simple pseudograph) with one edge which is a loop is a singleton of a singleton. (Contributed by AV, 17-Dec-2020.) (Revised by AV, 21-Feb-2021.)

Hypotheses

Ref	Expression
1loopgruspgr.v	⊢ (𝜑 → (Vtx‘𝐺) = 𝑉)
1loopgruspgr.a	⊢ (𝜑 → 𝐴 ∈ 𝑋)
1loopgruspgr.n	⊢ (𝜑 → 𝑁 ∈ 𝑉)
1loopgruspgr.i	⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})

Assertion

Ref	Expression
1loopgredg	⊢ (𝜑 → (Edg‘𝐺) = {{𝑁}})

Proof of Theorem 1loopgredg

Step	Hyp	Ref	Expression
1		edgval 29308	. . 3 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2	1	a1i 11	. 2 ⊢ (𝜑 → (Edg‘𝐺) = ran (iEdg‘𝐺))
3		1loopgruspgr.i	. . 3 ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})
4	3	rneqd 5919	. 2 ⊢ (𝜑 → ran (iEdg‘𝐺) = ran {⟨𝐴, {𝑁}⟩})
5		1loopgruspgr.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
6		rnsnopg 6212	. . 3 ⊢ (𝐴 ∈ 𝑋 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
7	5, 6	syl 18	. 2 ⊢ (𝜑 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
8	2, 4, 7	3eqtrd 2804	1 ⊢ (𝜑 → (Edg‘𝐺) = {{𝑁}})

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  {csn 4585  ⟨cop 4591  ran crn 5653  ‘cfv 6525  Vtxcvtx 29255  iEdgciedg 29256  Edgcedg 29306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-iota 6481  df-fun 6527  df-fv 6533  df-edg 29307

This theorem is referenced by:  1loopgrnb0  29761  1loopgrvd2  29762