Theorem lfgredgge2 29103

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 2735	. . . . 5 ⊢ 𝐴 = 𝐴
2		lfuhgrnloopv.e	. . . . 5 ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}
3	1, 2	feq23i 6700	. . . 4 ⊢ (𝐼:𝐴⟶𝐸 ↔ 𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
4	3	biimpi 216	. . 3 ⊢ (𝐼:𝐴⟶𝐸 → 𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
5	4	ffvelcdmda 7074	. 2 ⊢ ((𝐼:𝐴⟶𝐸 ∧ 𝑋 ∈ 𝐴) → (𝐼‘𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
6		fveq2 6876	. . . . 5 ⊢ (𝑦 = (𝐼‘𝑋) → (♯‘𝑦) = (♯‘(𝐼‘𝑋)))
7	6	breq2d 5131	. . . 4 ⊢ (𝑦 = (𝐼‘𝑋) → (2 ≤ (♯‘𝑦) ↔ 2 ≤ (♯‘(𝐼‘𝑋))))
8		fveq2 6876	. . . . . 6 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
9	8	breq2d 5131	. . . . 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 3674	. . 3 ⊢ ((𝐼‘𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ((𝐼‘𝑋) ∈ 𝒫 𝑉 ∧ 2 ≤ (♯‘(𝐼‘𝑋))))
12	11	simprbi 496	. 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 395   = wceq 1540   ∈ wcel 2108  {crab 3415  𝒫 cpw 4575   class class class wbr 5119  dom cdm 5654  ⟶wf 6527  ‘cfv 6531   ≤ cle 11270  2c2 12295  ♯chash 14348  iEdgciedg 28976

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539

This theorem is referenced by:  lfgrnloop  29104