Theorem grlicer 49207

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 49203	. 2 ⊢ (𝑓 ∈ UHGraph → 𝑓 ≃𝑙𝑔𝑟 𝑓)
2		grlicsym 49204	. 2 ⊢ (𝑓 ∈ UHGraph → (𝑓 ≃𝑙𝑔𝑟 𝑔 → 𝑔 ≃𝑙𝑔𝑟 𝑓))
3		grlictr 49206	. . 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 8750	1 ⊢ ( ≃𝑙𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2144   ∩ cin 4089   class class class wbr 5102   × cxp 5648   Er wer 8717  UHGraphcuhgr 29639   ≃_𝑙𝑔𝑟 cgrlic 49168

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-1o 8479  df-er 8720  df-map 8852  df-vtx 29581  df-iedg 29582  df-uhgr 29641  df-clnbgr 49010  df-isubgr 49052  df-grim 49069  df-gric 49072  df-grlim 49169  df-grlic 49172

This theorem is used by: (None)