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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1hegrlfgr | Structured version Visualization version GIF version |
Description: A graph 𝐺 with one hyperedge joining at least two vertices is a loop-free graph. (Contributed by AV, 23-Feb-2021.) |
Ref | Expression |
---|---|
1hegrlfgr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
1hegrlfgr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
1hegrlfgr.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
1hegrlfgr.n | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
1hegrlfgr.x | ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉) |
1hegrlfgr.i | ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩}) |
1hegrlfgr.e | ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝐸) |
Ref | Expression |
---|---|
1hegrlfgr | ⊢ (𝜑 → (iEdg‘𝐺):{𝐴}⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1hegrlfgr.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
2 | 1hegrlfgr.x | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉) | |
3 | f1osng 6868 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐸 ∈ 𝒫 𝑉) → {⟨𝐴, 𝐸⟩}:{𝐴}–1-1-onto→{𝐸}) | |
4 | 1, 2, 3 | syl2anc 583 | . . . 4 ⊢ (𝜑 → {⟨𝐴, 𝐸⟩}:{𝐴}–1-1-onto→{𝐸}) |
5 | f1of 6827 | . . . 4 ⊢ ({⟨𝐴, 𝐸⟩}:{𝐴}–1-1-onto→{𝐸} → {⟨𝐴, 𝐸⟩}:{𝐴}⟶{𝐸}) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → {⟨𝐴, 𝐸⟩}:{𝐴}⟶{𝐸}) |
7 | 1hegrlfgr.e | . . . . . . 7 ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝐸) | |
8 | 1hegrlfgr.b | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
9 | prid1g 4759 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵, 𝐶}) | |
10 | 8, 9 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ {𝐵, 𝐶}) |
11 | 7, 10 | sseldd 3978 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐸) |
12 | 1hegrlfgr.c | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
13 | prid2g 4760 | . . . . . . . 8 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ {𝐵, 𝐶}) | |
14 | 12, 13 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ {𝐵, 𝐶}) |
15 | 7, 14 | sseldd 3978 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐸) |
16 | 1hegrlfgr.n | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
17 | 2, 11, 15, 16 | nehash2 14441 | . . . . 5 ⊢ (𝜑 → 2 ≤ (♯‘𝐸)) |
18 | fveq2 6885 | . . . . . . 7 ⊢ (𝑥 = 𝐸 → (♯‘𝑥) = (♯‘𝐸)) | |
19 | 18 | breq2d 5153 | . . . . . 6 ⊢ (𝑥 = 𝐸 → (2 ≤ (♯‘𝑥) ↔ 2 ≤ (♯‘𝐸))) |
20 | 19 | elrab 3678 | . . . . 5 ⊢ (𝐸 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ (𝐸 ∈ 𝒫 𝑉 ∧ 2 ≤ (♯‘𝐸))) |
21 | 2, 17, 20 | sylanbrc 582 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
22 | 21 | snssd 4807 | . . 3 ⊢ (𝜑 → {𝐸} ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
23 | 6, 22 | fssd 6729 | . 2 ⊢ (𝜑 → {⟨𝐴, 𝐸⟩}:{𝐴}⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
24 | 1hegrlfgr.i | . . 3 ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩}) | |
25 | 24 | feq1d 6696 | . 2 ⊢ (𝜑 → ((iEdg‘𝐺):{𝐴}⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ {⟨𝐴, 𝐸⟩}:{𝐴}⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})) |
26 | 23, 25 | mpbird 257 | 1 ⊢ (𝜑 → (iEdg‘𝐺):{𝐴}⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 {crab 3426 ⊆ wss 3943 𝒫 cpw 4597 {csn 4623 {cpr 4625 ⟨cop 4629 class class class wbr 5141 ⟶wf 6533 –1-1-onto→wf1o 6536 ‘cfv 6537 ≤ cle 11253 2c2 12271 ♯chash 14295 iEdgciedg 28765 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-oadd 8471 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-fz 13491 df-hash 14296 |
This theorem is referenced by: (None) |
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