Theorem usgredg2ALT 28980

Description: Alternate proof of usgredg2 28979, not using umgredg2 28887. (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 16-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
usgredg2.e	⊢ 𝐸 = (iEdg‘𝐺)

Assertion

Ref	Expression
usgredg2ALT	⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (♯‘(𝐸‘𝑋)) = 2)

Proof of Theorem usgredg2ALT

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2727	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		usgredg2.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	usgrf 28942	. . . 4 ⊢ (𝐺 ∈ USGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
4		f1f 6787	. . . 4 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
5	3, 4	syl 17	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
6	5	ffvelcdmda 7088	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (𝐸‘𝑋) ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
7		fveq2 6891	. . . . 5 ⊢ (𝑥 = (𝐸‘𝑋) → (♯‘𝑥) = (♯‘(𝐸‘𝑋)))
8	7	eqeq1d 2729	. . . 4 ⊢ (𝑥 = (𝐸‘𝑋) → ((♯‘𝑥) = 2 ↔ (♯‘(𝐸‘𝑋)) = 2))
9	8	elrab 3680	. . 3 ⊢ ((𝐸‘𝑋) ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ ((𝐸‘𝑋) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘(𝐸‘𝑋)) = 2))
10	9	simprbi 496	. 2 ⊢ ((𝐸‘𝑋) ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} → (♯‘(𝐸‘𝑋)) = 2)
11	6, 10	syl 17	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (♯‘(𝐸‘𝑋)) = 2)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {crab 3427   ∖ cdif 3941  ∅c0 4318  𝒫 cpw 4598  {csn 4624  dom cdm 5672  ⟶wf 6538  –1-1→wf1 6539  ‘cfv 6542  2c2 12283  ♯chash 14307  Vtxcvtx 28783  iEdgciedg 28784  USGraphcusgr 28936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fv 6550  df-usgr 28938

This theorem is referenced by: (None)