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Mirrors > Home > MPE Home > Th. List > vdgrfival | Structured version Visualization version GIF version |
Description: The value of the vertex degree function (for graphs of finite size). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) |
Ref | Expression |
---|---|
vdgrfival | ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑈 ∈ 𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vdgrval 26423 | . 2 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑈 ∈ 𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}))) | |
2 | rabfi 8070 | . . . . . . . 8 ⊢ (𝐴 ∈ Fin → {𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∈ Fin) | |
3 | hashcl 13009 | . . . . . . . 8 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)} ∈ Fin → (#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) ∈ ℕ0) | |
4 | 2, 3 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) ∈ ℕ0) |
5 | 4 | nn0red 11229 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) ∈ ℝ) |
6 | rabfi 8070 | . . . . . . . 8 ⊢ (𝐴 ∈ Fin → {𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ∈ Fin) | |
7 | hashcl 13009 | . . . . . . . 8 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}} ∈ Fin → (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}) ∈ ℕ0) | |
8 | 6, 7 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}) ∈ ℕ0) |
9 | 8 | nn0red 11229 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}) ∈ ℝ) |
10 | 5, 9 | jca 553 | . . . . 5 ⊢ (𝐴 ∈ Fin → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) ∈ ℝ ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}) ∈ ℝ)) |
11 | 10 | 3ad2ant3 1077 | . . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ Fin) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) ∈ ℝ ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}) ∈ ℝ)) |
12 | 11 | adantr 480 | . . 3 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑈 ∈ 𝑉) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) ∈ ℝ ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}) ∈ ℝ)) |
13 | rexadd 11937 | . . 3 ⊢ (((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) ∈ ℝ ∧ (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}) ∈ ℝ) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}})) = ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}))) | |
14 | 12, 13 | syl 17 | . 2 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑈 ∈ 𝑉) → ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}})) = ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}))) |
15 | 1, 14 | eqtrd 2644 | 1 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑈 ∈ 𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {crab 2900 {csn 4125 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℝcr 9814 + caddc 9818 ℕ0cn0 11169 +𝑒 cxad 11820 #chash 12979 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-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-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 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-n0 11170 df-z 11255 df-uz 11564 df-xadd 11823 df-hash 12980 df-vdgr 26421 |
This theorem is referenced by: vdgrfiun 26429 |
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