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Theorem lfgredgge2 27904
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 2738 . . . . 5 𝐴 = 𝐴
2 lfuhgrnloopv.e . . . . 5 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}
31, 2feq23i 6660 . . . 4 (𝐼:𝐴𝐸𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
43biimpi 215 . . 3 (𝐼:𝐴𝐸𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
54ffvelcdmda 7032 . 2 ((𝐼:𝐴𝐸𝑋𝐴) → (𝐼𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
6 fveq2 6840 . . . . 5 (𝑦 = (𝐼𝑋) → (♯‘𝑦) = (♯‘(𝐼𝑋)))
76breq2d 5116 . . . 4 (𝑦 = (𝐼𝑋) → (2 ≤ (♯‘𝑦) ↔ 2 ≤ (♯‘(𝐼𝑋))))
8 fveq2 6840 . . . . . 6 (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
98breq2d 5116 . . . . 5 (𝑥 = 𝑦 → (2 ≤ (♯‘𝑥) ↔ 2 ≤ (♯‘𝑦)))
109cbvrabv 3416 . . . 4 {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} = {𝑦 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑦)}
117, 10elrab2 3647 . . 3 ((𝐼𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ((𝐼𝑋) ∈ 𝒫 𝑉 ∧ 2 ≤ (♯‘(𝐼𝑋))))
1211simprbi 498 . 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 397   = wceq 1542  wcel 2107  {crab 3406  𝒫 cpw 4559   class class class wbr 5104  dom cdm 5632  wf 6490  cfv 6494  cle 11149  2c2 12167  chash 14184  iEdgciedg 27777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-fv 6502
This theorem is referenced by:  lfgrnloop  27905
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