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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 6826 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐸 ∈ 𝒫 𝑉) → {⟨𝐴, 𝐸⟩}:{𝐴}–1-1-onto→{𝐸}) | |
4 | 1, 2, 3 | syl2anc 585 | . . . 4 ⊢ (𝜑 → {⟨𝐴, 𝐸⟩}:{𝐴}–1-1-onto→{𝐸}) |
5 | f1of 6785 | . . . 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 4722 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ {𝐵, 𝐶}) | |
10 | 8, 9 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ {𝐵, 𝐶}) |
11 | 7, 10 | sseldd 3946 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐸) |
12 | 1hegrlfgr.c | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
13 | prid2g 4723 | . . . . . . . 8 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ {𝐵, 𝐶}) | |
14 | 12, 13 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ {𝐵, 𝐶}) |
15 | 7, 14 | sseldd 3946 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐸) |
16 | 1hegrlfgr.n | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
17 | 2, 11, 15, 16 | nehash2 14379 | . . . . 5 ⊢ (𝜑 → 2 ≤ (♯‘𝐸)) |
18 | fveq2 6843 | . . . . . . 7 ⊢ (𝑥 = 𝐸 → (♯‘𝑥) = (♯‘𝐸)) | |
19 | 18 | breq2d 5118 | . . . . . 6 ⊢ (𝑥 = 𝐸 → (2 ≤ (♯‘𝑥) ↔ 2 ≤ (♯‘𝐸))) |
20 | 19 | elrab 3646 | . . . . 5 ⊢ (𝐸 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ (𝐸 ∈ 𝒫 𝑉 ∧ 2 ≤ (♯‘𝐸))) |
21 | 2, 17, 20 | sylanbrc 584 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
22 | 21 | snssd 4770 | . . 3 ⊢ (𝜑 → {𝐸} ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
23 | 6, 22 | fssd 6687 | . 2 ⊢ (𝜑 → {⟨𝐴, 𝐸⟩}:{𝐴}⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) |
24 | 1hegrlfgr.i | . . 3 ⊢ (𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩}) | |
25 | 24 | feq1d 6654 | . 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 1542 ∈ wcel 2107 ≠ wne 2940 {crab 3406 ⊆ wss 3911 𝒫 cpw 4561 {csn 4587 {cpr 4589 ⟨cop 4593 class class class wbr 5106 ⟶wf 6493 –1-1-onto→wf1o 6496 ‘cfv 6497 ≤ cle 11195 2c2 12213 ♯chash 14236 iEdgciedg 27990 |
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-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-oadd 8417 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-dju 9842 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-n0 12419 df-xnn0 12491 df-z 12505 df-uz 12769 df-fz 13431 df-hash 14237 |
This theorem is referenced by: (None) |
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