Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > strisomgrop | Structured version Visualization version GIF version |
Description: A graph represented as an extensible structure with vertices as base set and indexed edges is isomorphic to a hypergraph represented as ordered pair with the same vertices and edges. (Contributed by AV, 11-Nov-2022.) |
Ref | Expression |
---|---|
strisomgrop.g | ⊢ 𝐺 = 〈𝑉, 𝐸〉 |
strisomgrop.h | ⊢ 𝐻 = {〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉} |
Ref | Expression |
---|---|
strisomgrop | ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐺 IsomGr 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1128 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐺 ∈ UHGraph) | |
2 | strisomgrop.h | . . . 4 ⊢ 𝐻 = {〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉} | |
3 | prex 5323 | . . . 4 ⊢ {〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉} ∈ V | |
4 | 2, 3 | eqeltri 2906 | . . 3 ⊢ 𝐻 ∈ V |
5 | 4 | a1i 11 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐻 ∈ V) |
6 | opvtxfv 26716 | . . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
7 | 6 | 3adant1 1122 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) |
8 | strisomgrop.g | . . . . 5 ⊢ 𝐺 = 〈𝑉, 𝐸〉 | |
9 | 8 | fveq2i 6666 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘〈𝑉, 𝐸〉) |
10 | 9 | a1i 11 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘𝐺) = (Vtx‘〈𝑉, 𝐸〉)) |
11 | 2 | struct2grvtx 26739 | . . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘𝐻) = 𝑉) |
12 | 11 | 3adant1 1122 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘𝐻) = 𝑉) |
13 | 7, 10, 12 | 3eqtr4d 2863 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘𝐺) = (Vtx‘𝐻)) |
14 | opiedgfv 26719 | . . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
15 | 14 | 3adant1 1122 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) |
16 | 8 | fveq2i 6666 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘〈𝑉, 𝐸〉) |
17 | 16 | a1i 11 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘𝐺) = (iEdg‘〈𝑉, 𝐸〉)) |
18 | 2 | struct2griedg 26740 | . . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘𝐻) = 𝐸) |
19 | 18 | 3adant1 1122 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘𝐻) = 𝐸) |
20 | 15, 17, 19 | 3eqtr4d 2863 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘𝐺) = (iEdg‘𝐻)) |
21 | isomgreqve 43867 | . 2 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ V) ∧ ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻))) → 𝐺 IsomGr 𝐻) | |
22 | 1, 5, 13, 20, 21 | syl22anc 834 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐺 IsomGr 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {cpr 4559 〈cop 4563 class class class wbr 5057 ‘cfv 6348 ndxcnx 16468 Basecbs 16471 .efcedgf 26701 Vtxcvtx 26708 iEdgciedg 26709 UHGraphcuhgr 26768 IsomGr cisomgr 43861 |
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-fal 1541 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-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-hash 13679 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-edgf 26702 df-vtx 26710 df-iedg 26711 df-uhgr 26770 df-isomgr 43863 |
This theorem is referenced by: (None) |
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