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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 29077 | . . . 4 ⊢ (𝐺 ∈ 𝑋 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
| 4 | 3 | adantr 480 | . . 3 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
| 5 | 4 | adantr 480 | . 2 ⊢ (((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
| 6 | simpr 484 | . . . 4 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → 𝐸 = 𝐹) | |
| 7 | 6 | dmeqd 5916 | . . . 4 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → dom 𝐸 = dom 𝐹) |
| 8 | pweq 4614 | . . . . . 6 ⊢ (𝑉 = 𝑊 → 𝒫 𝑉 = 𝒫 𝑊) | |
| 9 | 8 | difeq1d 4125 | . . . . 5 ⊢ (𝑉 = 𝑊 → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅})) |
| 10 | 9 | adantr 480 | . . . 4 ⊢ ((𝑉 = 𝑊 ∧ 𝐸 = 𝐹) → (𝒫 𝑉 ∖ {∅}) = (𝒫 𝑊 ∖ {∅})) |
| 11 | 6, 7, 10 | feq123d 6725 | . . 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 29077 | . . . . 5 ⊢ (𝐻 ∈ 𝑌 → (𝐻 ∈ UHGraph ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅}))) |
| 15 | 14 | adantl 481 | . . . 4 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐻 ∈ UHGraph ↔ 𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅}))) |
| 16 | 15 | bicomd 223 | . . 3 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐹:dom 𝐹⟶(𝒫 𝑊 ∖ {∅}) ↔ 𝐻 ∈ UHGraph)) |
| 17 | 11, 16 | sylan9bbr 510 | . 2 ⊢ (((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ↔ 𝐻 ∈ UHGraph)) |
| 18 | 5, 17 | bitrd 279 | 1 ⊢ (((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐻 ∈ UHGraph)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ∅c0 4333 𝒫 cpw 4600 {csn 4626 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 Vtxcvtx 29013 iEdgciedg 29014 UHGraphcuhgr 29073 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-uhgr 29075 |
| This theorem is referenced by: (None) |
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