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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 2610 | . . . . 5 ⊢ 𝐴 = 𝐴 | |
2 | lfuhgrnloopv.e | . . . . 5 ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} | |
3 | 1, 2 | feq23i 5952 | . . . 4 ⊢ (𝐼:𝐴⟶𝐸 ↔ 𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) |
4 | 3 | biimpi 205 | . . 3 ⊢ (𝐼:𝐴⟶𝐸 → 𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) |
5 | 4 | ffvelrnda 6267 | . 2 ⊢ ((𝐼:𝐴⟶𝐸 ∧ 𝑋 ∈ 𝐴) → (𝐼‘𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) |
6 | fveq2 6103 | . . . . 5 ⊢ (𝑦 = (𝐼‘𝑋) → (#‘𝑦) = (#‘(𝐼‘𝑋))) | |
7 | 6 | breq2d 4595 | . . . 4 ⊢ (𝑦 = (𝐼‘𝑋) → (2 ≤ (#‘𝑦) ↔ 2 ≤ (#‘(𝐼‘𝑋)))) |
8 | fveq2 6103 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (#‘𝑥) = (#‘𝑦)) | |
9 | 8 | breq2d 4595 | . . . . 5 ⊢ (𝑥 = 𝑦 → (2 ≤ (#‘𝑥) ↔ 2 ≤ (#‘𝑦))) |
10 | 9 | cbvrabv 3172 | . . . 4 ⊢ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} = {𝑦 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑦)} |
11 | 7, 10 | elrab2 3333 | . . 3 ⊢ ((𝐼‘𝑋) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} ↔ ((𝐼‘𝑋) ∈ 𝒫 𝑉 ∧ 2 ≤ (#‘(𝐼‘𝑋)))) |
12 | 11 | simprbi 479 | . 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 383 = wceq 1475 ∈ wcel 1977 {crab 2900 𝒫 cpw 4108 class class class wbr 4583 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 ≤ cle 9954 2c2 10947 #chash 12979 iEdgciedg 25674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 |
This theorem is referenced by: lfgrnloop 25791 |
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