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Theorem lfgredgge2 29156
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:𝐴𝐸𝑋𝐴) → 2 ≤ (♯‘(𝐼𝑋)))
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.
StepHypRef Expression
1 eqid 2735 . . . . 5 𝐴 = 𝐴
2 lfuhgrnloopv.e . . . . 5 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}
31, 2feq23i 6731 . . . 4 (𝐼:𝐴𝐸𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
43biimpi 216 . . 3 (𝐼:𝐴𝐸𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
54ffvelcdmda 7104 . 2 ((𝐼:𝐴𝐸𝑋𝐴) → (𝐼𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
6 fveq2 6907 . . . . 5 (𝑦 = (𝐼𝑋) → (♯‘𝑦) = (♯‘(𝐼𝑋)))
76breq2d 5160 . . . 4 (𝑦 = (𝐼𝑋) → (2 ≤ (♯‘𝑦) ↔ 2 ≤ (♯‘(𝐼𝑋))))
8 fveq2 6907 . . . . . 6 (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
98breq2d 5160 . . . . 5 (𝑥 = 𝑦 → (2 ≤ (♯‘𝑥) ↔ 2 ≤ (♯‘𝑦)))
109cbvrabv 3444 . . . 4 {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} = {𝑦 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑦)}
117, 10elrab2 3698 . . 3 ((𝐼𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ((𝐼𝑋) ∈ 𝒫 𝑉 ∧ 2 ≤ (♯‘(𝐼𝑋))))
1211simprbi 496 . 2 ((𝐼𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → 2 ≤ (♯‘(𝐼𝑋)))
135, 12syl 17 1 ((𝐼:𝐴𝐸𝑋𝐴) → 2 ≤ (♯‘(𝐼𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {crab 3433  𝒫 cpw 4605   class class class wbr 5148  dom cdm 5689  wf 6559  cfv 6563  cle 11294  2c2 12319  chash 14366  iEdgciedg 29029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571
This theorem is referenced by:  lfgrnloop  29157
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