Theorem usgredg2ALT 29280

Description: Alternate proof of usgredg2 29279, not using umgredg2 29187. (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 2739	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		usgredg2.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	usgrf 29242	. . . 4 ⊢ (𝐺 ∈ USGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
4		f1f 6723	. . . 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 7025	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (𝐸‘𝑋) ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
7		fveq2 6827	. . . . 5 ⊢ (𝑥 = (𝐸‘𝑋) → (♯‘𝑥) = (♯‘(𝐸‘𝑋)))
8	7	eqeq1d 2741	. . . 4 ⊢ (𝑥 = (𝐸‘𝑋) → ((♯‘𝑥) = 2 ↔ (♯‘(𝐸‘𝑋)) = 2))
9	8	elrab 3629	. . 3 ⊢ ((𝐸‘𝑋) ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ ((𝐸‘𝑋) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘(𝐸‘𝑋)) = 2))
10	9	simprbi 498	. 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 396   = wceq 1547   ∈ wcel 2119  {crab 3391   ∖ cdif 3880  ∅c0 4261  𝒫 cpw 4529  {csn 4555  dom cdm 5618  ⟶wf 6481  –1-1→wf1 6482  ‘cfv 6485  2c2 12227  ♯chash 14283  Vtxcvtx 29083  iEdgciedg 29084  USGraphcusgr 29236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fv 6493  df-usgr 29238

This theorem is referenced by: (None)