Theorem lfgredg2dom 16056

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 ∣ 2o ≼ 𝑥}

Assertion

Ref	Expression
lfgredg2dom	⊢ ((𝐼:𝐴⟶𝐸 ∧ 𝑋 ∈ 𝐴) → 2o ≼ (𝐼‘𝑋))

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

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

Proof of Theorem lfgredg2dom

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2231	. . . . 5 ⊢ 𝐴 = 𝐴
2		lfuhgrnloopv.e	. . . . 5 ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2o ≼ 𝑥}
3	1, 2	feq23i 5484	. . . 4 ⊢ (𝐼:𝐴⟶𝐸 ↔ 𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2o ≼ 𝑥})
4	3	biimpi 120	. . 3 ⊢ (𝐼:𝐴⟶𝐸 → 𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2o ≼ 𝑥})
5	4	ffvelcdmda 5790	. 2 ⊢ ((𝐼:𝐴⟶𝐸 ∧ 𝑋 ∈ 𝐴) → (𝐼‘𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2o ≼ 𝑥})
6		breq2 4097	. . . 4 ⊢ (𝑦 = (𝐼‘𝑋) → (2o ≼ 𝑦 ↔ 2o ≼ (𝐼‘𝑋)))
7		breq2 4097	. . . . 5 ⊢ (𝑥 = 𝑦 → (2o ≼ 𝑥 ↔ 2o ≼ 𝑦))
8	7	cbvrabv 2802	. . . 4 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ 2o ≼ 𝑥} = {𝑦 ∈ 𝒫 𝑉 ∣ 2o ≼ 𝑦}
9	6, 8	elrab2 2966	. . 3 ⊢ ((𝐼‘𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2o ≼ 𝑥} ↔ ((𝐼‘𝑋) ∈ 𝒫 𝑉 ∧ 2o ≼ (𝐼‘𝑋)))
10	9	simprbi 275	. 2 ⊢ ((𝐼‘𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2o ≼ 𝑥} → 2o ≼ (𝐼‘𝑋))
11	5, 10	syl 14	1 ⊢ ((𝐼:𝐴⟶𝐸 ∧ 𝑋 ∈ 𝐴) → 2o ≼ (𝐼‘𝑋))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2202  {crab 2515  𝒫 cpw 3656   class class class wbr 4093  dom cdm 4731  ⟶wf 5329  ‘cfv 5333  2_oc2o 6619   ≼ cdom 6951  iEdgciedg 15937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341

This theorem is referenced by:  lfgrnloopen  16057