Theorem 1loopgredg 29587

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 29134	. . 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 5895	. 2 ⊢ (𝜑 → ran (iEdg‘𝐺) = ran {⟨𝐴, {𝑁}⟩})
5		1loopgruspgr.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
6		rnsnopg 6187	. . 3 ⊢ (𝐴 ∈ 𝑋 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
7	5, 6	syl 17	. 2 ⊢ (𝜑 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
8	2, 4, 7	3eqtrd 2776	1 ⊢ (𝜑 → (Edg‘𝐺) = {{𝑁}})

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {csn 4582  ⟨cop 4588  ran crn 5633  ‘cfv 6500  Vtxcvtx 29081  iEdgciedg 29082  Edgcedg 29132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fv 6508  df-edg 29133

This theorem is referenced by:  1loopgrnb0  29588  1loopgrvd2  29589