Theorem grlicsymb 47909

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 47908	. 2 ⊢ (𝐴 ∈ UHGraph → (𝐴 ≃𝑙𝑔𝑟 𝐵 → 𝐵 ≃𝑙𝑔𝑟 𝐴))
2		grlicsym 47908	. 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 2105   class class class wbr 5147  UHGraphcuhgr 29087   ≃_𝑙𝑔𝑟 cgrlic 47879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-1o 8504  df-map 8866  df-vtx 29029  df-iedg 29030  df-uhgr 29089  df-clnbgr 47743  df-isubgr 47784  df-grim 47801  df-gric 47804  df-grlim 47880  df-grlic 47883

This theorem is referenced by: (None)