![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2738 | . . . . 5 ⊢ 𝐴 = 𝐴 | |
2 | lfuhgrnloopv.e | . . . . 5 ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} | |
3 | 1, 2 | feq23i 6660 | . . . 4 ⊢ (𝐼:𝐴⟶𝐸 ↔ 𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
4 | 3 | biimpi 215 | . . 3 ⊢ (𝐼:𝐴⟶𝐸 → 𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
5 | 4 | ffvelcdmda 7032 | . 2 ⊢ ((𝐼:𝐴⟶𝐸 ∧ 𝑋 ∈ 𝐴) → (𝐼‘𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
6 | fveq2 6840 | . . . . 5 ⊢ (𝑦 = (𝐼‘𝑋) → (♯‘𝑦) = (♯‘(𝐼‘𝑋))) | |
7 | 6 | breq2d 5116 | . . . 4 ⊢ (𝑦 = (𝐼‘𝑋) → (2 ≤ (♯‘𝑦) ↔ 2 ≤ (♯‘(𝐼‘𝑋)))) |
8 | fveq2 6840 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦)) | |
9 | 8 | breq2d 5116 | . . . . 5 ⊢ (𝑥 = 𝑦 → (2 ≤ (♯‘𝑥) ↔ 2 ≤ (♯‘𝑦))) |
10 | 9 | cbvrabv 3416 | . . . 4 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} = {𝑦 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑦)} |
11 | 7, 10 | elrab2 3647 | . . 3 ⊢ ((𝐼‘𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ((𝐼‘𝑋) ∈ 𝒫 𝑉 ∧ 2 ≤ (♯‘(𝐼‘𝑋)))) |
12 | 11 | simprbi 498 | . 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 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 |
Copyright terms: Public domain | W3C validator |