Theorem vdusgraval 26434

Description: The value of the vertex degree function for simple undirected graphs. (Contributed by Alexander van der Vekens, 20-Dec-2017.)

Assertion

Ref	Expression
vdusgraval	⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}))

Distinct variable groups:   𝑥,𝐸   𝑥,𝑈   𝑥,𝑉

Proof of Theorem vdusgraval

Step	Hyp	Ref	Expression
1		usgranloop0 25909	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}} = ∅)
2		usgrafun 25878	. . . . . . 7 ⊢ (𝑉 USGrph 𝐸 → Fun 𝐸)
3		usgrav 25867	. . . . . . 7 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
4		funfn 5833	. . . . . . . 8 ⊢ (Fun 𝐸 ↔ 𝐸 Fn dom 𝐸)
5		simpl 472	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝑉 ∈ V)
6	5	adantl 481	. . . . . . . . . 10 ⊢ ((𝐸 Fn dom 𝐸 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V)) → 𝑉 ∈ V)
7		simpl 472	. . . . . . . . . 10 ⊢ ((𝐸 Fn dom 𝐸 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V)) → 𝐸 Fn dom 𝐸)
8		dmexg 6989	. . . . . . . . . . . 12 ⊢ (𝐸 ∈ V → dom 𝐸 ∈ V)
9	8	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → dom 𝐸 ∈ V)
10	9	adantl 481	. . . . . . . . . 10 ⊢ ((𝐸 Fn dom 𝐸 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V)) → dom 𝐸 ∈ V)
11	6, 7, 10	3jca 1235	. . . . . . . . 9 ⊢ ((𝐸 Fn dom 𝐸 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V)) → (𝑉 ∈ V ∧ 𝐸 Fn dom 𝐸 ∧ dom 𝐸 ∈ V))
12	11	ex 449	. . . . . . . 8 ⊢ (𝐸 Fn dom 𝐸 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ∈ V ∧ 𝐸 Fn dom 𝐸 ∧ dom 𝐸 ∈ V)))
13	4, 12	sylbi 206	. . . . . . 7 ⊢ (Fun 𝐸 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ∈ V ∧ 𝐸 Fn dom 𝐸 ∧ dom 𝐸 ∈ V)))
14	2, 3, 13	sylc 63	. . . . . 6 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 Fn dom 𝐸 ∧ dom 𝐸 ∈ V))
15	14	anim1i 590	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → ((𝑉 ∈ V ∧ 𝐸 Fn dom 𝐸 ∧ dom 𝐸 ∈ V) ∧ 𝑈 ∈ 𝑉))
16		vdgrval 26423	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 Fn dom 𝐸 ∧ dom 𝐸 ∈ V) ∧ 𝑈 ∈ 𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}})))
17		fveq2 6103	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}} = ∅ → (#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}}) = (#‘∅))
18		hash0 13019	. . . . . . . . . . 11 ⊢ (#‘∅) = 0
19	17, 18	syl6eq 2660	. . . . . . . . . 10 ⊢ ({𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}} = ∅ → (#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}}) = 0)
20		oveq2 6557	. . . . . . . . . . . . 13 ⊢ ((#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}}) = 0 → ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}})) = ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 0))
21	20	adantr 480	. . . . . . . . . . . 12 ⊢ (((#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}}) = 0 ∧ 𝑉 USGrph 𝐸) → ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}})) = ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 0))
22	3	simprd 478	. . . . . . . . . . . . . . 15 ⊢ (𝑉 USGrph 𝐸 → 𝐸 ∈ V)
23		rabexg 4739	. . . . . . . . . . . . . . 15 ⊢ (dom 𝐸 ∈ V → {𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∈ V)
24	22, 8, 23	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑉 USGrph 𝐸 → {𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∈ V)
25	24	adantl 481	. . . . . . . . . . . . 13 ⊢ (((#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}}) = 0 ∧ 𝑉 USGrph 𝐸) → {𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∈ V)
26		hashxrcl 13010	. . . . . . . . . . . . 13 ⊢ ({𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∈ V → (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) ∈ ℝ*)
27		xaddid1 11946	. . . . . . . . . . . . 13 ⊢ ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) ∈ ℝ* → ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 0) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}))
28	25, 26, 27	3syl 18	. . . . . . . . . . . 12 ⊢ (((#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}}) = 0 ∧ 𝑉 USGrph 𝐸) → ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 0) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}))
29	21, 28	eqtrd 2644	. . . . . . . . . . 11 ⊢ (((#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}}) = 0 ∧ 𝑉 USGrph 𝐸) → ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}})) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}))
30	29	ex 449	. . . . . . . . . 10 ⊢ ((#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}}) = 0 → (𝑉 USGrph 𝐸 → ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}})) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)})))
31	19, 30	syl 17	. . . . . . . . 9 ⊢ ({𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}} = ∅ → (𝑉 USGrph 𝐸 → ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}})) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)})))
32	31	com12 32	. . . . . . . 8 ⊢ (𝑉 USGrph 𝐸 → ({𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}} = ∅ → ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}})) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)})))
33	32	adantl 481	. . . . . . 7 ⊢ ((((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}})) ∧ 𝑉 USGrph 𝐸) → ({𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}} = ∅ → ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}})) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)})))
34		eqeq1 2614	. . . . . . . 8 ⊢ (((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}})) → (((𝑉 VDeg 𝐸)‘𝑈) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) ↔ ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}})) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)})))
35	34	adantr 480	. . . . . . 7 ⊢ ((((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}})) ∧ 𝑉 USGrph 𝐸) → (((𝑉 VDeg 𝐸)‘𝑈) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) ↔ ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}})) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)})))
36	33, 35	sylibrd 248	. . . . . 6 ⊢ ((((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}})) ∧ 𝑉 USGrph 𝐸) → ({𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}} = ∅ → ((𝑉 VDeg 𝐸)‘𝑈) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)})))
37	36	ex 449	. . . . 5 ⊢ (((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}})) → (𝑉 USGrph 𝐸 → ({𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}} = ∅ → ((𝑉 VDeg 𝐸)‘𝑈) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}))))
38	15, 16, 37	3syl 18	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → (𝑉 USGrph 𝐸 → ({𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}} = ∅ → ((𝑉 VDeg 𝐸)‘𝑈) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}))))
39	38	com12 32	. . 3 ⊢ (𝑉 USGrph 𝐸 → ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → ({𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}} = ∅ → ((𝑉 VDeg 𝐸)‘𝑈) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}))))
40	39	anabsi5 854	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → ({𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}} = ∅ → ((𝑉 VDeg 𝐸)‘𝑈) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)})))
41	1, 40	mpd 15	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ∅c0 3874  {csn 4125   class class class wbr 4583  dom cdm 5038  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ℝ^*cxr 9952   +_𝑒 cxad 11820  #chash 12979   USGrph cusg 25859   VDeg cvdg 26420

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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  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-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-hash 12980  df-usgra 25862  df-vdgr 26421

This theorem is referenced by:  vdusgra0nedg  26435  hashnbgravd  26439  hashnbgravdg  26440  usgravd0nedg  26445  vdn0frgrav2  26550  vdgn0frgrav2  26551  vdn1frgrav2  26552  vdgn1frgrav2  26553