Theorem 1loopgredg 29429

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 28976	. . 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 5902	. 2 ⊢ (𝜑 → ran (iEdg‘𝐺) = ran {⟨𝐴, {𝑁}⟩})
5		1loopgruspgr.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
6		rnsnopg 6194	. . 3 ⊢ (𝐴 ∈ 𝑋 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
7	5, 6	syl 17	. 2 ⊢ (𝜑 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
8	2, 4, 7	3eqtrd 2768	1 ⊢ (𝜑 → (Edg‘𝐺) = {{𝑁}})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {csn 4589  ⟨cop 4595  ran crn 5639  ‘cfv 6511  Vtxcvtx 28923  iEdgciedg 28924  Edgcedg 28974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fv 6519  df-edg 28975

This theorem is referenced by:  1loopgrnb0  29430  1loopgrvd2  29431