Theorem uspgrloopedg 29504

Description: The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29234) is a singleton of a singleton. (Contributed by AV, 17-Dec-2020.)

Hypothesis

Ref	Expression
uspgrloopvtx.g	⊢ 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩

Assertion

Ref	Expression
uspgrloopedg	⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘𝐺) = {{𝑁}})

Proof of Theorem uspgrloopedg

Step	Hyp	Ref	Expression
1		uspgrloopvtx.g	. . . 4 ⊢ 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩
2	1	fveq2i 6831	. . 3 ⊢ (Edg‘𝐺) = (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩)
3		snex 5376	. . . . 5 ⊢ {⟨𝐴, {𝑁}⟩} ∈ V
4	3	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑋 → {⟨𝐴, {𝑁}⟩} ∈ V)
5		edgopval 29036	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ {⟨𝐴, {𝑁}⟩} ∈ V) → (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = ran {⟨𝐴, {𝑁}⟩})
6	4, 5	sylan2 593	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = ran {⟨𝐴, {𝑁}⟩})
7	2, 6	eqtrid 2778	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘𝐺) = ran {⟨𝐴, {𝑁}⟩})
8		rnsnopg 6174	. . 3 ⊢ (𝐴 ∈ 𝑋 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
9	8	adantl 481	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
10	7, 9	eqtrd 2766	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘𝐺) = {{𝑁}})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436  {csn 4575  ⟨cop 4581  ran crn 5620  ‘cfv 6487  Edgcedg 29032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6443  df-fun 6489  df-fv 6495  df-2nd 7928  df-iedg 28984  df-edg 29033

This theorem is referenced by: (None)