![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6899 | . . 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 28631 | . . . . 5 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph) | |
5 | 4 | 3ad2ant2 1132 | . . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐺 ∈ UHGraph) |
6 | simp1 1134 | . . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑆 SubGraph 𝐺) | |
7 | simp3 1136 | . . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom 𝐼) | |
8 | 2, 3, 5, 6, 7 | subgruhgredgd 28808 | . . 3 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ (𝒫 𝑉 ∖ {∅})) |
9 | eqid 2730 | . . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
10 | 9 | uhgrfun 28593 | . . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺)) |
11 | 4, 10 | syl 17 | . . . . . . 7 ⊢ (𝐺 ∈ UMGraph → Fun (iEdg‘𝐺)) |
12 | 11 | 3ad2ant2 1132 | . . . . . 6 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → Fun (iEdg‘𝐺)) |
13 | eqid 2730 | . . . . . . . . 9 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆) | |
14 | eqid 2730 | . . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
15 | eqid 2730 | . . . . . . . . 9 ⊢ (Edg‘𝑆) = (Edg‘𝑆) | |
16 | 13, 14, 3, 9, 15 | subgrprop2 28798 | . . . . . . . 8 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))) |
17 | 16 | simp2d 1141 | . . . . . . 7 ⊢ (𝑆 SubGraph 𝐺 → 𝐼 ⊆ (iEdg‘𝐺)) |
18 | 17 | 3ad2ant1 1131 | . . . . . 6 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐼 ⊆ (iEdg‘𝐺)) |
19 | funssfv 6911 | . . . . . . 7 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → ((iEdg‘𝐺)‘𝑋) = (𝐼‘𝑋)) | |
20 | 19 | eqcomd 2736 | . . . . . 6 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝐼 ⊆ (iEdg‘𝐺) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) = ((iEdg‘𝐺)‘𝑋)) |
21 | 12, 18, 7, 20 | syl3anc 1369 | . . . . 5 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) = ((iEdg‘𝐺)‘𝑋)) |
22 | 21 | fveq2d 6894 | . . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) = (♯‘((iEdg‘𝐺)‘𝑋))) |
23 | simp2 1135 | . . . . 5 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐺 ∈ UMGraph) | |
24 | 3 | dmeqi 5903 | . . . . . . . . 9 ⊢ dom 𝐼 = dom (iEdg‘𝑆) |
25 | 24 | eleq2i 2823 | . . . . . . . 8 ⊢ (𝑋 ∈ dom 𝐼 ↔ 𝑋 ∈ dom (iEdg‘𝑆)) |
26 | subgreldmiedg 28807 | . . . . . . . . 9 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺)) | |
27 | 26 | ex 411 | . . . . . . . 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 1109 | . . . . 5 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom (iEdg‘𝐺)) |
31 | 14, 9 | umgredg2 28627 | . . . . 5 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom (iEdg‘𝐺)) → (♯‘((iEdg‘𝐺)‘𝑋)) = 2) |
32 | 23, 30, 31 | syl2anc 582 | . . . 4 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘((iEdg‘𝐺)‘𝑋)) = 2) |
33 | 22, 32 | eqtrd 2770 | . . 3 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) = 2) |
34 | 1, 8, 33 | elrabd 3684 | . 2 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ {𝑒 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑒) = 2}) |
35 | prprrab 14438 | . 2 ⊢ {𝑒 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑒) = 2} = {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} | |
36 | 34, 35 | eleqtrdi 2841 | 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 1085 = wceq 1539 ∈ wcel 2104 {crab 3430 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4321 𝒫 cpw 4601 {csn 4627 class class class wbr 5147 dom cdm 5675 Fun wfun 6536 ‘cfv 6542 2c2 12271 ♯chash 14294 Vtxcvtx 28523 iEdgciedg 28524 Edgcedg 28574 UHGraphcuhgr 28583 UMGraphcumgr 28608 SubGraph csubgr 28791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-hash 14295 df-edg 28575 df-uhgr 28585 df-upgr 28609 df-umgr 28610 df-subgr 28792 |
This theorem is referenced by: subumgr 28812 subusgr 28813 |
Copyright terms: Public domain | W3C validator |