Theorem lfgredg2dom 16578

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 2238	. . . . 5 ⊢ 𝐴 = 𝐴
2		lfuhgrnloopv.e	. . . . 5 ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2o ≼ 𝑥}
3	1, 2	feq23i 5531	. . . 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 5846	. 2 ⊢ ((𝐼:𝐴⟶𝐸 ∧ 𝑋 ∈ 𝐴) → (𝐼‘𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2o ≼ 𝑥})
6		breq2 4135	. . . 4 ⊢ (𝑦 = (𝐼‘𝑋) → (2o ≼ 𝑦 ↔ 2o ≼ (𝐼‘𝑋)))
7		breq2 4135	. . . . 5 ⊢ (𝑥 = 𝑦 → (2o ≼ 𝑥 ↔ 2o ≼ 𝑦))
8	7	cbvrabv 2820	. . . 4 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ 2o ≼ 𝑥} = {𝑦 ∈ 𝒫 𝑉 ∣ 2o ≼ 𝑦}
9	6, 8	elrab2 2985	. . 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  {crab 2532  𝒫 cpw 3688   class class class wbr 4131  dom cdm 4776  ⟶wf 5376  ‘cfv 5380  2_oc2o 6685   ≼ cdom 7025  iEdgciedg 16459

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388

This theorem is used by:  lfgrnloopen  16579