Theorem usgredg2ALT 29210

Description: Alternate proof of usgredg2 29209, not using umgredg2 29117. (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 2737	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		usgredg2.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	usgrf 29172	. . . 4 ⊢ (𝐺 ∈ USGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
4		f1f 6804	. . . 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 7104	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (𝐸‘𝑋) ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
7		fveq2 6906	. . . . 5 ⊢ (𝑥 = (𝐸‘𝑋) → (♯‘𝑥) = (♯‘(𝐸‘𝑋)))
8	7	eqeq1d 2739	. . . 4 ⊢ (𝑥 = (𝐸‘𝑋) → ((♯‘𝑥) = 2 ↔ (♯‘(𝐸‘𝑋)) = 2))
9	8	elrab 3692	. . 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 1540   ∈ wcel 2108  {crab 3436   ∖ cdif 3948  ∅c0 4333  𝒫 cpw 4600  {csn 4626  dom cdm 5685  ⟶wf 6557  –1-1→wf1 6558  ‘cfv 6561  2c2 12321  ♯chash 14369  Vtxcvtx 29013  iEdgciedg 29014  USGraphcusgr 29166

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569  df-usgr 29168

This theorem is referenced by: (None)