Theorem uspgrloopiedg 29452

Description: The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29183) 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 6864	. 2 ⊢ (iEdg‘𝐺) = (iEdg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩)
3		snex 5394	. . . 4 ⊢ {⟨𝐴, {𝑁}⟩} ∈ V
4	3	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑋 → {⟨𝐴, {𝑁}⟩} ∈ V)
5		opiedgfv 28941	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ {⟨𝐴, {𝑁}⟩} ∈ V) → (iEdg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = {⟨𝐴, {𝑁}⟩})
6	4, 5	sylan2 593	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (iEdg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = {⟨𝐴, {𝑁}⟩})
7	2, 6	eqtrid 2777	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  {csn 4592  ⟨cop 4598  ‘cfv 6514  iEdgciedg 28931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fv 6522  df-2nd 7972  df-iedg 28933

This theorem is referenced by:  uspgrloopnb0  29454  uspgrloopvd2  29455