Theorem uspgrloopiedg 30080

Description: The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29812) is a singleton of a singleton. (Contributed by AV, 21-Feb-2021.)

Hypothesis

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

Assertion

Ref	Expression
uspgrloopiedg	⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})

Proof of Theorem uspgrloopiedg

Step	Hyp	Ref	Expression
1		uspgrloopvtx.g	. . 3 ⊢ 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩
2	1	fveq2i 6880	. 2 ⊢ (iEdg‘𝐺) = (iEdg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩)
3		snex 5397	. . . 4 ⊢ {⟨𝐴, {𝑁}⟩} ∈ V
4	3	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑋 → {⟨𝐴, {𝑁}⟩} ∈ V)
5		opiedgfv 29567	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ {⟨𝐴, {𝑁}⟩} ∈ V) → (iEdg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = {⟨𝐴, {𝑁}⟩})
6	4, 5	sylan2 605	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (iEdg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = {⟨𝐴, {𝑁}⟩})
7	2, 6	eqtrid 2808	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  {csn 4584  ⟨cop 4590  ‘cfv 6531  iEdgciedg 29557

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 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7740

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6487  df-fun 6533  df-fv 6539  df-2nd 7991  df-iedg 29559

This theorem is used by:  uspgrloopnb0  30082  uspgrloopvd2  30083