Theorem usgredg2ALT 29282

Description: Alternate proof of usgredg2 29281, not using umgredg2 29189. (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 29244	. . . 4 ⊢ (𝐺 ∈ USGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
4		f1f 6738	. . . 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 7038	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (𝐸‘𝑋) ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
7		fveq2 6842	. . . . 5 ⊢ (𝑥 = (𝐸‘𝑋) → (♯‘𝑥) = (♯‘(𝐸‘𝑋)))
8	7	eqeq1d 2739	. . . 4 ⊢ (𝑥 = (𝐸‘𝑋) → ((♯‘𝑥) = 2 ↔ (♯‘(𝐸‘𝑋)) = 2))
9	8	elrab 3648	. . 3 ⊢ ((𝐸‘𝑋) ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ ((𝐸‘𝑋) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘(𝐸‘𝑋)) = 2))
10	9	simprbi 497	. 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 1542   ∈ wcel 2114  {crab 3401   ∖ cdif 3900  ∅c0 4287  𝒫 cpw 4556  {csn 4582  dom cdm 5632  ⟶wf 6496  –1-1→wf1 6497  ‘cfv 6500  2c2 12212  ♯chash 14265  Vtxcvtx 29085  iEdgciedg 29086  USGraphcusgr 29238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fv 6508  df-usgr 29240

This theorem is referenced by: (None)