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Mirrors > Home > MPE Home > Th. List > uhgreq12g | Structured version Visualization version GIF version |
Description: If two sets have the same vertices and the same edges, one set is a hypergraph iff the other set is a hypergraph. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.) |
Ref | Expression |
---|---|
uhgrf.v | ⊢ 𝑉 = (Vtx‘𝐺) |
uhgrf.e | ⊢ 𝐸 = (iEdg‘𝐺) |
uhgreq12g.w | ⊢ 𝑊 = (Vtx‘𝐻) |
uhgreq12g.f | ⊢ 𝐹 = (iEdg‘𝐻) |
Ref | Expression |
---|---|
uhgreq12g | ⊢ (((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐻 ∈ UHGraph)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgrf.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | uhgrf.e | . . . . 5 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 1, 2 | isuhgr 26839 | . . . 4 ⊢ (𝐺 ∈ 𝑋 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
4 | 3 | adantr 483 | . . 3 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
5 | 4 | adantr 483 | . 2 ⊢ (((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
6 | simpr 487 | . . . 4 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → 𝐸 = 𝐹) | |
7 | 6 | dmeqd 5769 | . . . 4 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → dom 𝐸 = dom 𝐹) |
8 | pweq 4542 | . . . . . 6 ⊢ (𝑉 = 𝑊 → 𝒫 𝑉 = 𝒫 𝑊) | |
9 | 8 | difeq1d 4098 | . . . . 5 ⊢ (𝑉 = 𝑊 → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅})) |
10 | 9 | adantr 483 | . . . 4 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅})) |
11 | 6, 7, 10 | feq123d 6498 | . . 3 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅}))) |
12 | uhgreq12g.w | . . . . . 6 ⊢ 𝑊 = (Vtx‘𝐻) | |
13 | uhgreq12g.f | . . . . . 6 ⊢ 𝐹 = (iEdg‘𝐻) | |
14 | 12, 13 | isuhgr 26839 | . . . . 5 ⊢ (𝐻 ∈ 𝑌 → (𝐻 ∈ UHGraph ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅}))) |
15 | 14 | adantl 484 | . . . 4 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐻 ∈ UHGraph ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅}))) |
16 | 15 | bicomd 225 | . . 3 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅}) ↔ 𝐻 ∈ UHGraph)) |
17 | 11, 16 | sylan9bbr 513 | . 2 ⊢ (((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐻 ∈ UHGraph)) |
18 | 5, 17 | bitrd 281 | 1 ⊢ (((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐻 ∈ UHGraph)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∖ cdif 3933 ∅c0 4291 𝒫 cpw 4539 {csn 4561 dom cdm 5550 ⟶wf 6346 ‘cfv 6350 Vtxcvtx 26775 iEdgciedg 26776 UHGraphcuhgr 26835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-nul 5203 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 df-uhgr 26837 |
This theorem is referenced by: (None) |
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