Theorem grlicer 47835

Description: Local isomorphism is an equivalence relation on hypergraphs. (Contributed by AV, 11-Jun-2025.)

Assertion

Ref	Expression
grlicer	⊢ ( ≃𝑙𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph

Proof of Theorem grlicer

Dummy variables 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grlicref 47831	. 2 ⊢ (𝑓 ∈ UHGraph → 𝑓 ≃𝑙𝑔𝑟 𝑓)
2		grlicsym 47832	. 2 ⊢ (𝑓 ∈ UHGraph → (𝑓 ≃𝑙𝑔𝑟 𝑔 → 𝑔 ≃𝑙𝑔𝑟 𝑓))
3		grlictr 47834	. . 3 ⊢ ((𝑓 ≃𝑙𝑔𝑟 𝑔 ∧ 𝑔 ≃𝑙𝑔𝑟 ℎ) → 𝑓 ≃𝑙𝑔𝑟 ℎ)
4	3	a1i 11	. 2 ⊢ (𝑓 ∈ UHGraph → ((𝑓 ≃𝑙𝑔𝑟 𝑔 ∧ 𝑔 ≃𝑙𝑔𝑟 ℎ) → 𝑓 ≃𝑙𝑔𝑟 ℎ))
5	1, 2, 4	brinxper 8794	1 ⊢ ( ≃𝑙𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108   ∩ cin 3975   class class class wbr 5166   × cxp 5698   Er wer 8762  UHGraphcuhgr 29093   ≃_𝑙𝑔𝑟 cgrlic 47803

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-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772

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-reu 3389  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 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-suc 6403  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-1st 8032  df-2nd 8033  df-1o 8524  df-er 8765  df-map 8888  df-vtx 29035  df-iedg 29036  df-uhgr 29095  df-clnbgr 47695  df-isubgr 47735  df-grim 47750  df-gric 47753  df-grlim 47804  df-grlic 47807

This theorem is referenced by: (None)