| Mathbox for Alexander van der Vekens |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > gricsym | Structured version Visualization version GIF version | ||
| Description: Graph isomorphism is symmetric for hypergraphs. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 3-May-2025.) |
| Ref | Expression |
|---|---|
| gricsym | ⊢ (𝐺 ∈ UHGraph → (𝐺 ≃𝑔𝑟 𝑆 → 𝑆 ≃𝑔𝑟 𝐺)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | brgric 47881 | . . 3 ⊢ (𝐺 ≃𝑔𝑟 𝑆 ↔ (𝐺 GraphIso 𝑆) ≠ ∅) | |
| 2 | n0 4353 | . . 3 ⊢ ((𝐺 GraphIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐺 GraphIso 𝑆)) | |
| 3 | 1, 2 | bitri 275 | . 2 ⊢ (𝐺 ≃𝑔𝑟 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝐺 GraphIso 𝑆)) |
| 4 | grimcnv 47879 | . . . 4 ⊢ (𝐺 ∈ UHGraph → (𝑓 ∈ (𝐺 GraphIso 𝑆) → ◡𝑓 ∈ (𝑆 GraphIso 𝐺))) | |
| 5 | brgrici 47882 | . . . 4 ⊢ (◡𝑓 ∈ (𝑆 GraphIso 𝐺) → 𝑆 ≃𝑔𝑟 𝐺) | |
| 6 | 4, 5 | syl6 35 | . . 3 ⊢ (𝐺 ∈ UHGraph → (𝑓 ∈ (𝐺 GraphIso 𝑆) → 𝑆 ≃𝑔𝑟 𝐺)) |
| 7 | 6 | exlimdv 1933 | . 2 ⊢ (𝐺 ∈ UHGraph → (∃𝑓 𝑓 ∈ (𝐺 GraphIso 𝑆) → 𝑆 ≃𝑔𝑟 𝐺)) |
| 8 | 3, 7 | biimtrid 242 | 1 ⊢ (𝐺 ∈ UHGraph → (𝐺 ≃𝑔𝑟 𝑆 → 𝑆 ≃𝑔𝑟 𝐺)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 class class class wbr 5143 ◡ccnv 5684 (class class class)co 7431 UHGraphcuhgr 29073 GraphIso cgrim 47861 ≃𝑔𝑟 cgric 47862 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 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-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-1o 8506 df-map 8868 df-uhgr 29075 df-grim 47864 df-gric 47867 |
| This theorem is referenced by: gricsymb 47891 gricer 47893 grlicsym 47973 clnbgr3stgrgrlic 47979 usgrexmpl12ngric 47997 |
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