Theorem grlicsymb 47821

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 47820	. 2 ⊢ (𝐴 ∈ UHGraph → (𝐴 ≃𝑙𝑔𝑟 𝐵 → 𝐵 ≃𝑙𝑔𝑟 𝐴))
2		grlicsym 47820	. 2 ⊢ (𝐵 ∈ UHGraph → (𝐵 ≃𝑙𝑔𝑟 𝐴 → 𝐴 ≃𝑙𝑔𝑟 𝐵))
3	1, 2	anbiim 640	1 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph) → (𝐴 ≃𝑙𝑔𝑟 𝐵 ↔ 𝐵 ≃𝑙𝑔𝑟 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108   class class class wbr 5166  UHGraphcuhgr 29083   ≃_𝑙𝑔𝑟 cgrlic 47791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7764

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5701  df-rel 5702  df-cnv 5703  df-co 5704  df-dm 5705  df-rn 5706  df-res 5707  df-ima 5708  df-suc 6396  df-iota 6520  df-fun 6570  df-fn 6571  df-f 6572  df-f1 6573  df-fo 6574  df-f1o 6575  df-fv 6576  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8024  df-2nd 8025  df-1o 8516  df-map 8880  df-vtx 29025  df-iedg 29026  df-uhgr 29085  df-clnbgr 47683  df-isubgr 47723  df-grim 47738  df-gric 47741  df-grlim 47792  df-grlic 47795

This theorem is referenced by: (None)