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Mirrors > Home > MPE Home > Th. List > subumgredg2 | Structured version Visualization version GIF version |
Description: An edge of a subgraph of a multigraph connects exactly two different vertices. (Contributed by AV, 26-Nov-2020.) |
Ref | Expression |
---|---|
subumgredg2.v | ⊢ 𝑉 = (Vtx‘𝑆) |
subumgredg2.i | ⊢ 𝐼 = (iEdg‘𝑆) |
Ref | Expression |
---|---|
subumgredg2 | ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveqeq2 6901 | . . 3 ⊢ (𝑒 = (𝐼‘𝑋) → ((♯‘𝑒) = 2 ↔ (♯‘(𝐼‘𝑋)) = 2)) | |
2 | subumgredg2.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝑆) | |
3 | subumgredg2.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝑆) | |
4 | umgruhgr 28364 | . . . . 5 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph) | |
5 | 4 | 3ad2ant2 1135 | . . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐺 ∈ UHGraph) |
6 | simp1 1137 | . . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑆 SubGraph 𝐺) | |
7 | simp3 1139 | . . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom 𝐼) | |
8 | 2, 3, 5, 6, 7 | subgruhgredgd 28541 | . . 3 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ (𝒫 𝑉 ∖ {∅})) |
9 | eqid 2733 | . . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
10 | 9 | uhgrfun 28326 | . . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺)) |
11 | 4, 10 | syl 17 | . . . . . . 7 ⊢ (𝐺 ∈ UMGraph → Fun (iEdg‘𝐺)) |
12 | 11 | 3ad2ant2 1135 | . . . . . 6 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → Fun (iEdg‘𝐺)) |
13 | eqid 2733 | . . . . . . . . 9 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆) | |
14 | eqid 2733 | . . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
15 | eqid 2733 | . . . . . . . . 9 ⊢ (Edg‘𝑆) = (Edg‘𝑆) | |
16 | 13, 14, 3, 9, 15 | subgrprop2 28531 | . . . . . . . 8 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))) |
17 | 16 | simp2d 1144 | . . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → 𝐼 ⊆ (iEdg‘𝐺)) |
18 | 17 | 3ad2ant1 1134 | . . . . . 6 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐼 ⊆ (iEdg‘𝐺)) |
19 | funssfv 6913 | . . . . . . 7 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → ((iEdg‘𝐺)‘𝑋) = (𝐼‘𝑋)) | |
20 | 19 | eqcomd 2739 | . . . . . 6 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) = ((iEdg‘𝐺)‘𝑋)) |
21 | 12, 18, 7, 20 | syl3anc 1372 | . . . . 5 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) = ((iEdg‘𝐺)‘𝑋)) |
22 | 21 | fveq2d 6896 | . . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) = (♯‘((iEdg‘𝐺)‘𝑋))) |
23 | simp2 1138 | . . . . 5 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐺 ∈ UMGraph) | |
24 | 3 | dmeqi 5905 | . . . . . . . . 9 ⊢ dom 𝐼 = dom (iEdg‘𝑆) |
25 | 24 | eleq2i 2826 | . . . . . . . 8 ⊢ (𝑋 ∈ dom 𝐼 ↔ 𝑋 ∈ dom (iEdg‘𝑆)) |
26 | subgreldmiedg 28540 | . . . . . . . . 9 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺)) | |
27 | 26 | ex 414 | . . . . . . . 8 ⊢ (𝑆 SubGraph 𝐺 → (𝑋 ∈ dom (iEdg‘𝑆) → 𝑋 ∈ dom (iEdg‘𝐺))) |
28 | 25, 27 | biimtrid 241 | . . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → (𝑋 ∈ dom 𝐼 → 𝑋 ∈ dom (iEdg‘𝐺))) |
29 | 28 | a1d 25 | . . . . . 6 ⊢ (𝑆 SubGraph 𝐺 → (𝐺 ∈ UMGraph → (𝑋 ∈ dom 𝐼 → 𝑋 ∈ dom (iEdg‘𝐺)))) |
30 | 29 | 3imp 1112 | . . . . 5 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom (iEdg‘𝐺)) |
31 | 14, 9 | umgredg2 28360 | . . . . 5 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom (iEdg‘𝐺)) → (♯‘((iEdg‘𝐺)‘𝑋)) = 2) |
32 | 23, 30, 31 | syl2anc 585 | . . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘((iEdg‘𝐺)‘𝑋)) = 2) |
33 | 22, 32 | eqtrd 2773 | . . 3 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) = 2) |
34 | 1, 8, 33 | elrabd 3686 | . 2 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ {𝑒 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑒) = 2}) |
35 | prprrab 14434 | . 2 ⊢ {𝑒 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑒) = 2} = {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} | |
36 | 34, 35 | eleqtrdi 2844 | 1 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 {crab 3433 ∖ cdif 3946 ⊆ wss 3949 ∅c0 4323 𝒫 cpw 4603 {csn 4629 class class class wbr 5149 dom cdm 5677 Fun wfun 6538 ‘cfv 6544 2c2 12267 ♯chash 14290 Vtxcvtx 28256 iEdgciedg 28257 Edgcedg 28307 UHGraphcuhgr 28316 UMGraphcumgr 28341 SubGraph csubgr 28524 |
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 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-hash 14291 df-edg 28308 df-uhgr 28318 df-upgr 28342 df-umgr 28343 df-subgr 28525 |
This theorem is referenced by: subumgr 28545 subusgr 28546 |
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