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Mirrors > Home > MPE Home > Th. List > lfgredgge2 | Structured version Visualization version GIF version |
Description: An edge of a loop-free graph has at least two ends. (Contributed by AV, 23-Feb-2021.) |
Ref | Expression |
---|---|
lfuhgrnloopv.i | ⊢ 𝐼 = (iEdg‘𝐺) |
lfuhgrnloopv.a | ⊢ 𝐴 = dom 𝐼 |
lfuhgrnloopv.e | ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} |
Ref | Expression |
---|---|
lfgredgge2 | ⊢ ((𝐼:𝐴⟶𝐸 ∧ 𝑋 ∈ 𝐴) → 2 ≤ (♯‘(𝐼‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . . . 5 ⊢ 𝐴 = 𝐴 | |
2 | lfuhgrnloopv.e | . . . . 5 ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} | |
3 | 1, 2 | feq23i 6731 | . . . 4 ⊢ (𝐼:𝐴⟶𝐸 ↔ 𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
4 | 3 | biimpi 216 | . . 3 ⊢ (𝐼:𝐴⟶𝐸 → 𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
5 | 4 | ffvelcdmda 7104 | . 2 ⊢ ((𝐼:𝐴⟶𝐸 ∧ 𝑋 ∈ 𝐴) → (𝐼‘𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
6 | fveq2 6907 | . . . . 5 ⊢ (𝑦 = (𝐼‘𝑋) → (♯‘𝑦) = (♯‘(𝐼‘𝑋))) | |
7 | 6 | breq2d 5160 | . . . 4 ⊢ (𝑦 = (𝐼‘𝑋) → (2 ≤ (♯‘𝑦) ↔ 2 ≤ (♯‘(𝐼‘𝑋)))) |
8 | fveq2 6907 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦)) | |
9 | 8 | breq2d 5160 | . . . . 5 ⊢ (𝑥 = 𝑦 → (2 ≤ (♯‘𝑥) ↔ 2 ≤ (♯‘𝑦))) |
10 | 9 | cbvrabv 3444 | . . . 4 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} = {𝑦 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑦)} |
11 | 7, 10 | elrab2 3698 | . . 3 ⊢ ((𝐼‘𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ((𝐼‘𝑋) ∈ 𝒫 𝑉 ∧ 2 ≤ (♯‘(𝐼‘𝑋)))) |
12 | 11 | simprbi 496 | . 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))) |
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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