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Mirrors > Home > MPE Home > Th. List > uhgrstrrepe | Structured version Visualization version GIF version |
Description: Replacing (or adding) the edges (between elements of the base set) of an extensible structure results in a hypergraph. Instead of requiring (𝜑 → 𝐺 Struct 𝑋), it would be sufficient to require (𝜑 → Fun (𝐺 ∖ {∅})) and (𝜑 → 𝐺 ∈ V). (Contributed by AV, 18-Jan-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.) |
Ref | Expression |
---|---|
uhgrstrrepe.v | ⊢ 𝑉 = (Base‘𝐺) |
uhgrstrrepe.i | ⊢ 𝐼 = (.ef‘ndx) |
uhgrstrrepe.s | ⊢ (𝜑 → 𝐺 Struct 𝑋) |
uhgrstrrepe.b | ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺) |
uhgrstrrepe.w | ⊢ (𝜑 → 𝐸 ∈ 𝑊) |
uhgrstrrepe.e | ⊢ (𝜑 → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})) |
Ref | Expression |
---|---|
uhgrstrrepe | ⊢ (𝜑 → (𝐺 sSet 〈𝐼, 𝐸〉) ∈ UHGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgrstrrepe.e | . . . 4 ⊢ (𝜑 → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})) | |
2 | uhgrstrrepe.i | . . . . . . . . 9 ⊢ 𝐼 = (.ef‘ndx) | |
3 | uhgrstrrepe.s | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 Struct 𝑋) | |
4 | uhgrstrrepe.b | . . . . . . . . 9 ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺) | |
5 | uhgrstrrepe.w | . . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝑊) | |
6 | 2, 3, 4, 5 | setsvtx 26820 | . . . . . . . 8 ⊢ (𝜑 → (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (Base‘𝐺)) |
7 | uhgrstrrepe.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝐺) | |
8 | 6, 7 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝜑 → (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = 𝑉) |
9 | 8 | pweqd 4558 | . . . . . 6 ⊢ (𝜑 → 𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = 𝒫 𝑉) |
10 | 9 | difeq1d 4098 | . . . . 5 ⊢ (𝜑 → (𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∖ {∅}) = (𝒫 𝑉 ∖ {∅})) |
11 | 10 | feq3d 6501 | . . . 4 ⊢ (𝜑 → (𝐸:dom 𝐸⟶(𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∖ {∅}) ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
12 | 1, 11 | mpbird 259 | . . 3 ⊢ (𝜑 → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∖ {∅})) |
13 | 2, 3, 4, 5 | setsiedg 26821 | . . . 4 ⊢ (𝜑 → (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) = 𝐸) |
14 | 13 | dmeqd 5774 | . . . 4 ⊢ (𝜑 → dom (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) = dom 𝐸) |
15 | 13, 14 | feq12d 6502 | . . 3 ⊢ (𝜑 → ((iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)):dom (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉))⟶(𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∖ {∅}) ↔ 𝐸:dom 𝐸⟶(𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∖ {∅}))) |
16 | 12, 15 | mpbird 259 | . 2 ⊢ (𝜑 → (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)):dom (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉))⟶(𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∖ {∅})) |
17 | ovex 7189 | . . 3 ⊢ (𝐺 sSet 〈𝐼, 𝐸〉) ∈ V | |
18 | eqid 2821 | . . . 4 ⊢ (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) | |
19 | eqid 2821 | . . . 4 ⊢ (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) | |
20 | 18, 19 | isuhgr 26845 | . . 3 ⊢ ((𝐺 sSet 〈𝐼, 𝐸〉) ∈ V → ((𝐺 sSet 〈𝐼, 𝐸〉) ∈ UHGraph ↔ (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)):dom (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉))⟶(𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∖ {∅}))) |
21 | 17, 20 | mp1i 13 | . 2 ⊢ (𝜑 → ((𝐺 sSet 〈𝐼, 𝐸〉) ∈ UHGraph ↔ (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)):dom (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉))⟶(𝒫 (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) ∖ {∅}))) |
22 | 16, 21 | mpbird 259 | 1 ⊢ (𝜑 → (𝐺 sSet 〈𝐼, 𝐸〉) ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∖ cdif 3933 ∅c0 4291 𝒫 cpw 4539 {csn 4567 〈cop 4573 class class class wbr 5066 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 Struct cstr 16479 ndxcnx 16480 sSet csts 16481 Basecbs 16483 .efcedgf 26774 Vtxcvtx 26781 iEdgciedg 26782 UHGraphcuhgr 26841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-edgf 26775 df-vtx 26783 df-iedg 26784 df-uhgr 26843 |
This theorem is referenced by: (None) |
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