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Theorem grlicer 47753
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 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grlicref 47749 . 2 (𝑓 ∈ UHGraph → 𝑓𝑙𝑔𝑟 𝑓)
2 grlicsym 47750 . 2 (𝑓 ∈ UHGraph → (𝑓𝑙𝑔𝑟 𝑔𝑔𝑙𝑔𝑟 𝑓))
3 grlictr 47752 . . 3 ((𝑓𝑙𝑔𝑟 𝑔𝑔𝑙𝑔𝑟 ) → 𝑓𝑙𝑔𝑟 )
43a1i 11 . 2 (𝑓 ∈ UHGraph → ((𝑓𝑙𝑔𝑟 𝑔𝑔𝑙𝑔𝑟 ) → 𝑓𝑙𝑔𝑟 ))
51, 2, 4brinxper 8788 1 ( ≃𝑙𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2103  cin 3969   class class class wbr 5169   × cxp 5697   Er wer 8756  UHGraphcuhgr 29082  𝑙𝑔𝑟 cgrlic 47730
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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
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 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450  df-1st 8026  df-2nd 8027  df-1o 8518  df-er 8759  df-map 8882  df-vtx 29024  df-iedg 29025  df-uhgr 29084  df-clnbgr 47626  df-isubgr 47663  df-grim 47678  df-gric 47681  df-grlim 47731  df-grlic 47734
This theorem is referenced by: (None)
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