Theorem uspgrloopedg 29721

Description: The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29452) 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 6872	. . 3 ⊢ (Edg‘𝐺) = (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩)
3		snex 5398	. . . . 5 ⊢ {⟨𝐴, {𝑁}⟩} ∈ V
4	3	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑋 → {⟨𝐴, {𝑁}⟩} ∈ V)
5		edgopval 29254	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ {⟨𝐴, {𝑁}⟩} ∈ V) → (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = ran {⟨𝐴, {𝑁}⟩})
6	4, 5	sylan2 602	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = ran {⟨𝐴, {𝑁}⟩})
7	2, 6	eqtrid 2811	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘𝐺) = ran {⟨𝐴, {𝑁}⟩})
8		rnsnopg 6210	. . 3 ⊢ (𝐴 ∈ 𝑋 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
9	8	adantl 485	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
10	7, 9	eqtrd 2799	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘𝐺) = {{𝑁}})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  Vcvv 3456  {csn 4584  ⟨cop 4590  ran crn 5650  ‘cfv 6523  Edgcedg 29250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-iota 6479  df-fun 6525  df-fv 6531  df-2nd 7973  df-iedg 29202  df-edg 29251

This theorem is referenced by: (None)