Theorem lfgredgge2 27494

Description: An edge of a loop-free graph has at least two ends. (Contributed by AV, 23-Feb-2021.)

Hypotheses

Ref	Expression
lfuhgrnloopv.i	⊢ 𝐼 = (iEdg‘𝐺)
lfuhgrnloopv.a	⊢ 𝐴 = dom 𝐼
lfuhgrnloopv.e	⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}

Assertion

Ref	Expression
lfgredgge2	⊢ ((𝐼:𝐴⟶𝐸 ∧ 𝑋 ∈ 𝐴) → 2 ≤ (♯‘(𝐼‘𝑋)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐼   𝑥,𝑉

Allowed substitution hints:   𝐸(𝑥)   𝐺(𝑥)   𝑋(𝑥)

Proof of Theorem lfgredgge2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . . . 5 ⊢ 𝐴 = 𝐴
2		lfuhgrnloopv.e	. . . . 5 ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}
3	1, 2	feq23i 6594	. . . 4 ⊢ (𝐼:𝐴⟶𝐸 ↔ 𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
4	3	biimpi 215	. . 3 ⊢ (𝐼:𝐴⟶𝐸 → 𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
5	4	ffvelrnda 6961	. 2 ⊢ ((𝐼:𝐴⟶𝐸 ∧ 𝑋 ∈ 𝐴) → (𝐼‘𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
6		fveq2 6774	. . . . 5 ⊢ (𝑦 = (𝐼‘𝑋) → (♯‘𝑦) = (♯‘(𝐼‘𝑋)))
7	6	breq2d 5086	. . . 4 ⊢ (𝑦 = (𝐼‘𝑋) → (2 ≤ (♯‘𝑦) ↔ 2 ≤ (♯‘(𝐼‘𝑋))))
8		fveq2 6774	. . . . . 6 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
9	8	breq2d 5086	. . . . 5 ⊢ (𝑥 = 𝑦 → (2 ≤ (♯‘𝑥) ↔ 2 ≤ (♯‘𝑦)))
10	9	cbvrabv 3426	. . . 4 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} = {𝑦 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑦)}
11	7, 10	elrab2 3627	. . 3 ⊢ ((𝐼‘𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ((𝐼‘𝑋) ∈ 𝒫 𝑉 ∧ 2 ≤ (♯‘(𝐼‘𝑋))))
12	11	simprbi 497	. 2 ⊢ ((𝐼‘𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → 2 ≤ (♯‘(𝐼‘𝑋)))
13	5, 12	syl 17	1 ⊢ ((𝐼:𝐴⟶𝐸 ∧ 𝑋 ∈ 𝐴) → 2 ≤ (♯‘(𝐼‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {crab 3068  𝒫 cpw 4533   class class class wbr 5074  dom cdm 5589  ⟶wf 6429  ‘cfv 6433   ≤ cle 11010  2c2 12028  ♯chash 14044  iEdgciedg 27367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441

This theorem is referenced by:  lfgrnloop  27495