Theorem uspgrloopiedg 29403

Description: The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29134) 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 6899	. 2 ⊢ (iEdg‘𝐺) = (iEdg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩)
3		snex 5433	. . . 4 ⊢ {⟨𝐴, {𝑁}⟩} ∈ V
4	3	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑋 → {⟨𝐴, {𝑁}⟩} ∈ V)
5		opiedgfv 28892	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ {⟨𝐴, {𝑁}⟩} ∈ V) → (iEdg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = {⟨𝐴, {𝑁}⟩})
6	4, 5	sylan2 591	. 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 394   = wceq 1533   ∈ wcel 2098  Vcvv 3461  {csn 4630  ⟨cop 4636  ‘cfv 6549  iEdgciedg 28882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6501  df-fun 6551  df-fv 6557  df-2nd 7995  df-iedg 28884

This theorem is referenced by:  uspgrloopnb0  29405  uspgrloopvd2  29406