Theorem grlicsymb 48141

Description: Graph local isomorphism is symmetric in both directions for hypergraphs. (Contributed by AV, 9-Jun-2025.)

Assertion

Ref	Expression
grlicsymb	⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph) → (𝐴 ≃𝑙𝑔𝑟 𝐵 ↔ 𝐵 ≃𝑙𝑔𝑟 𝐴))

Proof of Theorem grlicsymb

Step	Hyp	Ref	Expression
1		grlicsym 48140	. 2 ⊢ (𝐴 ∈ UHGraph → (𝐴 ≃𝑙𝑔𝑟 𝐵 → 𝐵 ≃𝑙𝑔𝑟 𝐴))
2		grlicsym 48140	. 2 ⊢ (𝐵 ∈ UHGraph → (𝐵 ≃𝑙𝑔𝑟 𝐴 → 𝐴 ≃𝑙𝑔𝑟 𝐵))
3	1, 2	anbiim 641	1 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph) → (𝐴 ≃𝑙𝑔𝑟 𝐵 ↔ 𝐵 ≃𝑙𝑔𝑟 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2113   class class class wbr 5095  UHGraphcuhgr 29038   ≃_𝑙𝑔𝑟 cgrlic 48104

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7357  df-oprab 7358  df-mpo 7359  df-1st 7929  df-2nd 7930  df-1o 8393  df-map 8760  df-vtx 28980  df-iedg 28981  df-uhgr 29040  df-clnbgr 47946  df-isubgr 47988  df-grim 48005  df-gric 48008  df-grlim 48105  df-grlic 48108

This theorem is referenced by: (None)