Theorem grlicer 48504

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 48500	. 2 ⊢ (𝑓 ∈ UHGraph → 𝑓 ≃𝑙𝑔𝑟 𝑓)
2		grlicsym 48501	. 2 ⊢ (𝑓 ∈ UHGraph → (𝑓 ≃𝑙𝑔𝑟 𝑔 → 𝑔 ≃𝑙𝑔𝑟 𝑓))
3		grlictr 48503	. . 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 8666	1 ⊢ ( ≃𝑙𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114   ∩ cin 3889   class class class wbr 5086   × cxp 5622   Er wer 8633  UHGraphcuhgr 29139   ≃_𝑙𝑔𝑟 cgrlic 48465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-1o 8398  df-er 8636  df-map 8768  df-vtx 29081  df-iedg 29082  df-uhgr 29141  df-clnbgr 48307  df-isubgr 48349  df-grim 48366  df-gric 48369  df-grlim 48466  df-grlic 48469

This theorem is referenced by: (None)