Theorem usgredg2ALT 29543

Description: Alternate proof of usgredg2 29542, not using umgredg2 29450. (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 2763	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		usgredg2.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	usgrf 29505	. . . 4 ⊢ (𝐺 ∈ USGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
4		f1f 6774	. . . 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 18	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
6	5	ffvelcdmda 7079	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (𝐸‘𝑋) ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
7		fveq2 6881	. . . . 5 ⊢ (𝑥 = (𝐸‘𝑋) → (♯‘𝑥) = (♯‘(𝐸‘𝑋)))
8	7	eqeq1d 2765	. . . 4 ⊢ (𝑥 = (𝐸‘𝑋) → ((♯‘𝑥) = 2 ↔ (♯‘(𝐸‘𝑋)) = 2))
9	8	elrab 3650	. . 3 ⊢ ((𝐸‘𝑋) ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ ((𝐸‘𝑋) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘(𝐸‘𝑋)) = 2))
10	9	simprbi 502	. 2 ⊢ ((𝐸‘𝑋) ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} → (♯‘(𝐸‘𝑋)) = 2)
11	6, 10	syl 18	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ dom 𝐸) → (♯‘(𝐸‘𝑋)) = 2)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {crab 3416   ∖ cdif 3902  ∅c0 4286  𝒫 cpw 4562  {csn 4589  dom cdm 5661  ⟶wf 6532  –1-1→wf1 6533  ‘cfv 6536  2c2 12290  ♯chash 14362  Vtxcvtx 29346  iEdgciedg 29347  USGraphcusgr 29499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fv 6544  df-usgr 29501

This theorem is referenced by: (None)