Theorem 1loopgredg 29962

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 29507	. . 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 5922	. 2 ⊢ (𝜑 → ran (iEdg‘𝐺) = ran {⟨𝐴, {𝑁}⟩})
5		1loopgruspgr.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
6		rnsnopg 6217	. . 3 ⊢ (𝐴 ∈ 𝑋 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
7	5, 6	syl 18	. 2 ⊢ (𝜑 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
8	2, 4, 7	3eqtrd 2799	1 ⊢ (𝜑 → (Edg‘𝐺) = {{𝑁}})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {csn 4584  ⟨cop 4590  ran crn 5656  ‘cfv 6533  Vtxcvtx 29454  iEdgciedg 29455  Edgcedg 29505

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fv 6541  df-edg 29506

This theorem is used by:  1loopgrnb0  29963  1loopgrvd2  29964