Theorem grlicer 48370

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 48366	. 2 ⊢ (𝑓 ∈ UHGraph → 𝑓 ≃𝑙𝑔𝑟 𝑓)
2		grlicsym 48367	. 2 ⊢ (𝑓 ∈ UHGraph → (𝑓 ≃𝑙𝑔𝑟 𝑔 → 𝑔 ≃𝑙𝑔𝑟 𝑓))
3		grlictr 48369	. . 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 8675	1 ⊢ ( ≃𝑙𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114   ∩ cin 3902   class class class wbr 5100   × cxp 5630   Er wer 8642  UHGraphcuhgr 29141   ≃_𝑙𝑔𝑟 cgrlic 48331

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-1o 8407  df-er 8645  df-map 8777  df-vtx 29083  df-iedg 29084  df-uhgr 29143  df-clnbgr 48173  df-isubgr 48215  df-grim 48232  df-gric 48235  df-grlim 48332  df-grlic 48335

This theorem is referenced by: (None)