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Theorem lfgredgge2 29141
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 2737 . . . . 5 𝐴 = 𝐴
2 lfuhgrnloopv.e . . . . 5 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}
31, 2feq23i 6730 . . . 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 6906 . . . . 5 (𝑦 = (𝐼𝑋) → (♯‘𝑦) = (♯‘(𝐼𝑋)))
76breq2d 5155 . . . 4 (𝑦 = (𝐼𝑋) → (2 ≤ (♯‘𝑦) ↔ 2 ≤ (♯‘(𝐼𝑋))))
8 fveq2 6906 . . . . . 6 (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
98breq2d 5155 . . . . 5 (𝑥 = 𝑦 → (2 ≤ (♯‘𝑥) ↔ 2 ≤ (♯‘𝑦)))
109cbvrabv 3447 . . . 4 {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} = {𝑦 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑦)}
117, 10elrab2 3695 . . 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 1540  wcel 2108  {crab 3436  𝒫 cpw 4600   class class class wbr 5143  dom cdm 5685  wf 6557  cfv 6561  cle 11296  2c2 12321  chash 14369  iEdgciedg 29014
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 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by:  lfgrnloop  29142
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