Theorem lfgredg2dom 15924

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 2229	. . . . 5 ⊢ 𝐴 = 𝐴
2		lfuhgrnloopv.e	. . . . 5 ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2o ≼ 𝑥}
3	1, 2	feq23i 5467	. . . 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 5769	. 2 ⊢ ((𝐼:𝐴⟶𝐸 ∧ 𝑋 ∈ 𝐴) → (𝐼‘𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2o ≼ 𝑥})
6		breq2 4086	. . . 4 ⊢ (𝑦 = (𝐼‘𝑋) → (2o ≼ 𝑦 ↔ 2o ≼ (𝐼‘𝑋)))
7		breq2 4086	. . . . 5 ⊢ (𝑥 = 𝑦 → (2o ≼ 𝑥 ↔ 2o ≼ 𝑦))
8	7	cbvrabv 2798	. . . 4 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ 2o ≼ 𝑥} = {𝑦 ∈ 𝒫 𝑉 ∣ 2o ≼ 𝑦}
9	6, 8	elrab2 2962	. . 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 1395   ∈ wcel 2200  {crab 2512  𝒫 cpw 3649   class class class wbr 4082  dom cdm 4718  ⟶wf 5313  ‘cfv 5317  2_oc2o 6554   ≼ cdom 6884  iEdgciedg 15808

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325

This theorem is referenced by:  lfgrnloopen  15925