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Mirrors > Home > MPE Home > Th. List > p1evtxdeqlem | Structured version Visualization version GIF version |
Description: Lemma for p1evtxdeq 27222 and p1evtxdp1 27223. (Contributed by AV, 3-Mar-2021.) |
Ref | Expression |
---|---|
p1evtxdeq.v | ⊢ 𝑉 = (Vtx‘𝐺) |
p1evtxdeq.i | ⊢ 𝐼 = (iEdg‘𝐺) |
p1evtxdeq.f | ⊢ (𝜑 → Fun 𝐼) |
p1evtxdeq.fv | ⊢ (𝜑 → (Vtx‘𝐹) = 𝑉) |
p1evtxdeq.fi | ⊢ (𝜑 → (iEdg‘𝐹) = (𝐼 ∪ {〈𝐾, 𝐸〉})) |
p1evtxdeq.k | ⊢ (𝜑 → 𝐾 ∈ 𝑋) |
p1evtxdeq.d | ⊢ (𝜑 → 𝐾 ∉ dom 𝐼) |
p1evtxdeq.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
p1evtxdeq.e | ⊢ (𝜑 → 𝐸 ∈ 𝑌) |
Ref | Expression |
---|---|
p1evtxdeqlem | ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 ((VtxDeg‘〈𝑉, {〈𝐾, 𝐸〉}〉)‘𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | p1evtxdeq.i | . 2 ⊢ 𝐼 = (iEdg‘𝐺) | |
2 | p1evtxdeq.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | fvexi 6677 | . . . 4 ⊢ 𝑉 ∈ V |
4 | snex 5322 | . . . 4 ⊢ {〈𝐾, 𝐸〉} ∈ V | |
5 | 3, 4 | pm3.2i 471 | . . 3 ⊢ (𝑉 ∈ V ∧ {〈𝐾, 𝐸〉} ∈ V) |
6 | opiedgfv 26719 | . . . 4 ⊢ ((𝑉 ∈ V ∧ {〈𝐾, 𝐸〉} ∈ V) → (iEdg‘〈𝑉, {〈𝐾, 𝐸〉}〉) = {〈𝐾, 𝐸〉}) | |
7 | 6 | eqcomd 2824 | . . 3 ⊢ ((𝑉 ∈ V ∧ {〈𝐾, 𝐸〉} ∈ V) → {〈𝐾, 𝐸〉} = (iEdg‘〈𝑉, {〈𝐾, 𝐸〉}〉)) |
8 | 5, 7 | ax-mp 5 | . 2 ⊢ {〈𝐾, 𝐸〉} = (iEdg‘〈𝑉, {〈𝐾, 𝐸〉}〉) |
9 | opvtxfv 26716 | . . 3 ⊢ ((𝑉 ∈ V ∧ {〈𝐾, 𝐸〉} ∈ V) → (Vtx‘〈𝑉, {〈𝐾, 𝐸〉}〉) = 𝑉) | |
10 | 5, 9 | mp1i 13 | . 2 ⊢ (𝜑 → (Vtx‘〈𝑉, {〈𝐾, 𝐸〉}〉) = 𝑉) |
11 | p1evtxdeq.fv | . 2 ⊢ (𝜑 → (Vtx‘𝐹) = 𝑉) | |
12 | p1evtxdeq.e | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑌) | |
13 | dmsnopg 6063 | . . . . 5 ⊢ (𝐸 ∈ 𝑌 → dom {〈𝐾, 𝐸〉} = {𝐾}) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝜑 → dom {〈𝐾, 𝐸〉} = {𝐾}) |
15 | 14 | ineq2d 4186 | . . 3 ⊢ (𝜑 → (dom 𝐼 ∩ dom {〈𝐾, 𝐸〉}) = (dom 𝐼 ∩ {𝐾})) |
16 | p1evtxdeq.d | . . . . 5 ⊢ (𝜑 → 𝐾 ∉ dom 𝐼) | |
17 | df-nel 3121 | . . . . 5 ⊢ (𝐾 ∉ dom 𝐼 ↔ ¬ 𝐾 ∈ dom 𝐼) | |
18 | 16, 17 | sylib 219 | . . . 4 ⊢ (𝜑 → ¬ 𝐾 ∈ dom 𝐼) |
19 | disjsn 4639 | . . . 4 ⊢ ((dom 𝐼 ∩ {𝐾}) = ∅ ↔ ¬ 𝐾 ∈ dom 𝐼) | |
20 | 18, 19 | sylibr 235 | . . 3 ⊢ (𝜑 → (dom 𝐼 ∩ {𝐾}) = ∅) |
21 | 15, 20 | eqtrd 2853 | . 2 ⊢ (𝜑 → (dom 𝐼 ∩ dom {〈𝐾, 𝐸〉}) = ∅) |
22 | p1evtxdeq.f | . 2 ⊢ (𝜑 → Fun 𝐼) | |
23 | p1evtxdeq.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑋) | |
24 | funsng 6398 | . . 3 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → Fun {〈𝐾, 𝐸〉}) | |
25 | 23, 12, 24 | syl2anc 584 | . 2 ⊢ (𝜑 → Fun {〈𝐾, 𝐸〉}) |
26 | p1evtxdeq.u | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
27 | p1evtxdeq.fi | . 2 ⊢ (𝜑 → (iEdg‘𝐹) = (𝐼 ∪ {〈𝐾, 𝐸〉})) | |
28 | 1, 8, 2, 10, 11, 21, 22, 25, 26, 27 | vtxdun 27190 | 1 ⊢ (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 ((VtxDeg‘〈𝑉, {〈𝐾, 𝐸〉}〉)‘𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∉ wnel 3120 Vcvv 3492 ∪ cun 3931 ∩ cin 3932 ∅c0 4288 {csn 4557 〈cop 4563 dom cdm 5548 Fun wfun 6342 ‘cfv 6348 (class class class)co 7145 +𝑒 cxad 12493 Vtxcvtx 26708 iEdgciedg 26709 VtxDegcvtxdg 27174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-xadd 12496 df-hash 13679 df-vtx 26710 df-iedg 26711 df-vtxdg 27175 |
This theorem is referenced by: p1evtxdeq 27222 p1evtxdp1 27223 |
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