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Theorem 1hevtxdg1 26383
Description: The vertex degree of vertex 𝐷 in a graph 𝐺 with only one hyperedge 𝐸 (not being a loop) is 1 if 𝐷 is incident with the edge 𝐸. (Contributed by AV, 2-Mar-2021.) (Proof shortened by AV, 17-Apr-2021.)
Hypotheses
Ref Expression
1hevtxdg0.i (𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})
1hevtxdg0.v (𝜑 → (Vtx‘𝐺) = 𝑉)
1hevtxdg0.a (𝜑𝐴𝑋)
1hevtxdg0.d (𝜑𝐷𝑉)
1hevtxdg1.e (𝜑𝐸 ∈ 𝒫 𝑉)
1hevtxdg1.n (𝜑𝐷𝐸)
1hevtxdg1.l (𝜑 → 2 ≤ (#‘𝐸))
Assertion
Ref Expression
1hevtxdg1 (𝜑 → ((VtxDeg‘𝐺)‘𝐷) = 1)

Proof of Theorem 1hevtxdg1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 1hevtxdg0.i . . . 4 (𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})
21dmeqd 5315 . . 3 (𝜑 → dom (iEdg‘𝐺) = dom {⟨𝐴, 𝐸⟩})
3 1hevtxdg1.e . . . 4 (𝜑𝐸 ∈ 𝒫 𝑉)
4 dmsnopg 5594 . . . 4 (𝐸 ∈ 𝒫 𝑉 → dom {⟨𝐴, 𝐸⟩} = {𝐴})
53, 4syl 17 . . 3 (𝜑 → dom {⟨𝐴, 𝐸⟩} = {𝐴})
62, 5eqtrd 2654 . 2 (𝜑 → dom (iEdg‘𝐺) = {𝐴})
7 1hevtxdg0.a . . . . . . 7 (𝜑𝐴𝑋)
8 1hevtxdg0.v . . . . . . . . . 10 (𝜑 → (Vtx‘𝐺) = 𝑉)
98pweqd 4154 . . . . . . . . 9 (𝜑 → 𝒫 (Vtx‘𝐺) = 𝒫 𝑉)
103, 9eleqtrrd 2702 . . . . . . . 8 (𝜑𝐸 ∈ 𝒫 (Vtx‘𝐺))
11 1hevtxdg1.l . . . . . . . 8 (𝜑 → 2 ≤ (#‘𝐸))
12 fveq2 6178 . . . . . . . . . 10 (𝑥 = 𝐸 → (#‘𝑥) = (#‘𝐸))
1312breq2d 4656 . . . . . . . . 9 (𝑥 = 𝐸 → (2 ≤ (#‘𝑥) ↔ 2 ≤ (#‘𝐸)))
1413elrab 3357 . . . . . . . 8 (𝐸 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (#‘𝑥)} ↔ (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 2 ≤ (#‘𝐸)))
1510, 11, 14sylanbrc 697 . . . . . . 7 (𝜑𝐸 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (#‘𝑥)})
167, 15fsnd 6166 . . . . . 6 (𝜑 → {⟨𝐴, 𝐸⟩}:{𝐴}⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (#‘𝑥)})
1716adantr 481 . . . . 5 ((𝜑 ∧ dom (iEdg‘𝐺) = {𝐴}) → {⟨𝐴, 𝐸⟩}:{𝐴}⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (#‘𝑥)})
181adantr 481 . . . . . 6 ((𝜑 ∧ dom (iEdg‘𝐺) = {𝐴}) → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})
19 simpr 477 . . . . . 6 ((𝜑 ∧ dom (iEdg‘𝐺) = {𝐴}) → dom (iEdg‘𝐺) = {𝐴})
2018, 19feq12d 6020 . . . . 5 ((𝜑 ∧ dom (iEdg‘𝐺) = {𝐴}) → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (#‘𝑥)} ↔ {⟨𝐴, 𝐸⟩}:{𝐴}⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (#‘𝑥)}))
2117, 20mpbird 247 . . . 4 ((𝜑 ∧ dom (iEdg‘𝐺) = {𝐴}) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (#‘𝑥)})
22 1hevtxdg0.d . . . . . 6 (𝜑𝐷𝑉)
2322, 8eleqtrrd 2702 . . . . 5 (𝜑𝐷 ∈ (Vtx‘𝐺))
2423adantr 481 . . . 4 ((𝜑 ∧ dom (iEdg‘𝐺) = {𝐴}) → 𝐷 ∈ (Vtx‘𝐺))
25 eqid 2620 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
26 eqid 2620 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
27 eqid 2620 . . . . 5 dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
28 eqid 2620 . . . . 5 (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
2925, 26, 27, 28vtxdlfgrval 26362 . . . 4 (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (#‘𝑥)} ∧ 𝐷 ∈ (Vtx‘𝐺)) → ((VtxDeg‘𝐺)‘𝐷) = (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥)}))
3021, 24, 29syl2anc 692 . . 3 ((𝜑 ∧ dom (iEdg‘𝐺) = {𝐴}) → ((VtxDeg‘𝐺)‘𝐷) = (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥)}))
31 rabeq 3187 . . . . 5 (dom (iEdg‘𝐺) = {𝐴} → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ {𝐴} ∣ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥)})
3231adantl 482 . . . 4 ((𝜑 ∧ dom (iEdg‘𝐺) = {𝐴}) → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ {𝐴} ∣ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥)})
3332fveq2d 6182 . . 3 ((𝜑 ∧ dom (iEdg‘𝐺) = {𝐴}) → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥)}) = (#‘{𝑥 ∈ {𝐴} ∣ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥)}))
34 fveq2 6178 . . . . . . . . 9 (𝑥 = 𝐴 → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝐺)‘𝐴))
3534eleq2d 2685 . . . . . . . 8 (𝑥 = 𝐴 → (𝐷 ∈ ((iEdg‘𝐺)‘𝑥) ↔ 𝐷 ∈ ((iEdg‘𝐺)‘𝐴)))
3635rabsnif 4249 . . . . . . 7 {𝑥 ∈ {𝐴} ∣ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥)} = if(𝐷 ∈ ((iEdg‘𝐺)‘𝐴), {𝐴}, ∅)
37 1hevtxdg1.n . . . . . . . . 9 (𝜑𝐷𝐸)
381fveq1d 6180 . . . . . . . . . 10 (𝜑 → ((iEdg‘𝐺)‘𝐴) = ({⟨𝐴, 𝐸⟩}‘𝐴))
39 fvsng 6432 . . . . . . . . . . 11 ((𝐴𝑋𝐸 ∈ 𝒫 𝑉) → ({⟨𝐴, 𝐸⟩}‘𝐴) = 𝐸)
407, 3, 39syl2anc 692 . . . . . . . . . 10 (𝜑 → ({⟨𝐴, 𝐸⟩}‘𝐴) = 𝐸)
4138, 40eqtrd 2654 . . . . . . . . 9 (𝜑 → ((iEdg‘𝐺)‘𝐴) = 𝐸)
4237, 41eleqtrrd 2702 . . . . . . . 8 (𝜑𝐷 ∈ ((iEdg‘𝐺)‘𝐴))
4342iftrued 4085 . . . . . . 7 (𝜑 → if(𝐷 ∈ ((iEdg‘𝐺)‘𝐴), {𝐴}, ∅) = {𝐴})
4436, 43syl5eq 2666 . . . . . 6 (𝜑 → {𝑥 ∈ {𝐴} ∣ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝐴})
4544fveq2d 6182 . . . . 5 (𝜑 → (#‘{𝑥 ∈ {𝐴} ∣ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥)}) = (#‘{𝐴}))
46 hashsng 13142 . . . . . 6 (𝐴𝑋 → (#‘{𝐴}) = 1)
477, 46syl 17 . . . . 5 (𝜑 → (#‘{𝐴}) = 1)
4845, 47eqtrd 2654 . . . 4 (𝜑 → (#‘{𝑥 ∈ {𝐴} ∣ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥)}) = 1)
4948adantr 481 . . 3 ((𝜑 ∧ dom (iEdg‘𝐺) = {𝐴}) → (#‘{𝑥 ∈ {𝐴} ∣ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥)}) = 1)
5030, 33, 493eqtrd 2658 . 2 ((𝜑 ∧ dom (iEdg‘𝐺) = {𝐴}) → ((VtxDeg‘𝐺)‘𝐷) = 1)
516, 50mpdan 701 1 (𝜑 → ((VtxDeg‘𝐺)‘𝐷) = 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  {crab 2913  c0 3907  ifcif 4077  𝒫 cpw 4149  {csn 4168  cop 4174   class class class wbr 4644  dom cdm 5104  wf 5872  cfv 5876  1c1 9922  cle 10060  2c2 11055  #chash 13100  Vtxcvtx 25855  iEdgciedg 25856  VtxDegcvtxdg 26342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-n0 11278  df-xnn0 11349  df-z 11363  df-uz 11673  df-xadd 11932  df-fz 12312  df-hash 13101  df-vtxdg 26343
This theorem is referenced by:  1hegrvtxdg1  26384  p1evtxdp1  26391
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