Theorem uspgrloopedg 29962

Description: The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29693) 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 6885	. . 3 ⊢ (Edg‘𝐺) = (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩)
3		snex 5408	. . . . 5 ⊢ {⟨𝐴, {𝑁}⟩} ∈ V
4	3	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑋 → {⟨𝐴, {𝑁}⟩} ∈ V)
5		edgopval 29492	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ {⟨𝐴, {𝑁}⟩} ∈ V) → (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = ran {⟨𝐴, {𝑁}⟩})
6	4, 5	sylan2 605	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = ran {⟨𝐴, {𝑁}⟩})
7	2, 6	eqtrid 2809	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘𝐺) = ran {⟨𝐴, {𝑁}⟩})
8		rnsnopg 6221	. . 3 ⊢ (𝐴 ∈ 𝑋 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
9	8	adantl 487	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
10	7, 9	eqtrd 2797	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘𝐺) = {{𝑁}})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453  {csn 4587  ⟨cop 4593  ran crn 5660  ‘cfv 6537  Edgcedg 29488

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fv 6545  df-2nd 7990  df-iedg 29440  df-edg 29489

This theorem is used by: (None)