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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 47819 | . . 3 ⊢ (𝐺 ≃𝑔𝑟 𝑆 ↔ (𝐺 GraphIso 𝑆) ≠ ∅) | |
2 | n0 4359 | . . 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 47817 | . . . 4 ⊢ (𝐺 ∈ UHGraph → (𝑓 ∈ (𝐺 GraphIso 𝑆) → ◡𝑓 ∈ (𝑆 GraphIso 𝐺))) | |
5 | brgrici 47820 | . . . 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 1931 | . 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 1776 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 class class class wbr 5148 ◡ccnv 5688 (class class class)co 7431 UHGraphcuhgr 29088 GraphIso cgrim 47799 ≃𝑔𝑟 cgric 47800 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-1o 8505 df-map 8867 df-uhgr 29090 df-grim 47802 df-gric 47805 |
This theorem is referenced by: gricsymb 47829 gricer 47831 grlicsym 47909 clnbgr3stgrgrlic 47915 usgrexmpl12ngric 47933 |
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