Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  lfgredgge2 Structured version   Visualization version   GIF version

Theorem lfgredgge2 26240
 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 2771 . . . . 5 𝐴 = 𝐴
2 lfuhgrnloopv.e . . . . 5 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}
31, 2feq23i 6178 . . . 4 (𝐼:𝐴𝐸𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
43biimpi 206 . . 3 (𝐼:𝐴𝐸𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
54ffvelrnda 6504 . 2 ((𝐼:𝐴𝐸𝑋𝐴) → (𝐼𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
6 fveq2 6333 . . . . 5 (𝑦 = (𝐼𝑋) → (♯‘𝑦) = (♯‘(𝐼𝑋)))
76breq2d 4799 . . . 4 (𝑦 = (𝐼𝑋) → (2 ≤ (♯‘𝑦) ↔ 2 ≤ (♯‘(𝐼𝑋))))
8 fveq2 6333 . . . . . 6 (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
98breq2d 4799 . . . . 5 (𝑥 = 𝑦 → (2 ≤ (♯‘𝑥) ↔ 2 ≤ (♯‘𝑦)))
109cbvrabv 3349 . . . 4 {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} = {𝑦 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑦)}
117, 10elrab2 3518 . . 3 ((𝐼𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ((𝐼𝑋) ∈ 𝒫 𝑉 ∧ 2 ≤ (♯‘(𝐼𝑋))))
1211simprbi 484 . 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 382   = wceq 1631   ∈ wcel 2145  {crab 3065  𝒫 cpw 4298   class class class wbr 4787  dom cdm 5250  ⟶wf 6026  ‘cfv 6030   ≤ cle 10281  2c2 11276  ♯chash 13321  iEdgciedg 26096 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038 This theorem is referenced by:  lfgrnloop  26241
 Copyright terms: Public domain W3C validator