Theorem lfgredgge2 26019

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:𝐴⟶𝐸 ∧ 𝑋 ∈ 𝐴) → 2 ≤ (#‘(𝐼‘𝑋)))

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.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 ⊢ 𝐴 = 𝐴
2		lfuhgrnloopv.e	. . . . 5 ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}
3	1, 2	feq23i 6039	. . . 4 ⊢ (𝐼:𝐴⟶𝐸 ↔ 𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)})
4	3	biimpi 206	. . 3 ⊢ (𝐼:𝐴⟶𝐸 → 𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)})
5	4	ffvelrnda 6359	. 2 ⊢ ((𝐼:𝐴⟶𝐸 ∧ 𝑋 ∈ 𝐴) → (𝐼‘𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)})
6		fveq2 6191	. . . . 5 ⊢ (𝑦 = (𝐼‘𝑋) → (#‘𝑦) = (#‘(𝐼‘𝑋)))
7	6	breq2d 4665	. . . 4 ⊢ (𝑦 = (𝐼‘𝑋) → (2 ≤ (#‘𝑦) ↔ 2 ≤ (#‘(𝐼‘𝑋))))
8		fveq2 6191	. . . . . 6 ⊢ (𝑥 = 𝑦 → (#‘𝑥) = (#‘𝑦))
9	8	breq2d 4665	. . . . 5 ⊢ (𝑥 = 𝑦 → (2 ≤ (#‘𝑥) ↔ 2 ≤ (#‘𝑦)))
10	9	cbvrabv 3199	. . . 4 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = {𝑦 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑦)}
11	7, 10	elrab2 3366	. . 3 ⊢ ((𝐼‘𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} ↔ ((𝐼‘𝑋) ∈ 𝒫 𝑉 ∧ 2 ≤ (#‘(𝐼‘𝑋))))
12	11	simprbi 480	. 2 ⊢ ((𝐼‘𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} → 2 ≤ (#‘(𝐼‘𝑋)))
13	5, 12	syl 17	1 ⊢ ((𝐼:𝐴⟶𝐸 ∧ 𝑋 ∈ 𝐴) → 2 ≤ (#‘(𝐼‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916  𝒫 cpw 4158   class class class wbr 4653  dom cdm 5114  ⟶wf 5884  ‘cfv 5888   ≤ cle 10075  2c2 11070  #chash 13117  iEdgciedg 25875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  lfgrnloop  26020