Theorem 1loopgredg 29545

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 29092	. . 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 5956	. 2 ⊢ (𝜑 → ran (iEdg‘𝐺) = ran {⟨𝐴, {𝑁}⟩})
5		1loopgruspgr.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
6		rnsnopg 6249	. . 3 ⊢ (𝐴 ∈ 𝑋 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
7	5, 6	syl 17	. 2 ⊢ (𝜑 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
8	2, 4, 7	3eqtrd 2781	1 ⊢ (𝜑 → (Edg‘𝐺) = {{𝑁}})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  {csn 4634  ⟨cop 4640  ran crn 5694  ‘cfv 6569  Vtxcvtx 29039  iEdgciedg 29040  Edgcedg 29090

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-iota 6522  df-fun 6571  df-fv 6577  df-edg 29091

This theorem is referenced by:  1loopgrnb0  29546  1loopgrvd2  29547