Theorem gricer 47831

Description: Isomorphism is an equivalence relation on hypergraphs. (Contributed by AV, 3-May-2025.) (Proof shortened by AV, 11-Jul-2025.)

Assertion

Ref	Expression
gricer	⊢ ( ≃𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph

Proof of Theorem gricer

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

Step	Hyp	Ref	Expression
1		gricref 47827	. 2 ⊢ (𝑔 ∈ UHGraph → 𝑔 ≃𝑔𝑟 𝑔)
2		gricsym 47828	. 2 ⊢ (𝑔 ∈ UHGraph → (𝑔 ≃𝑔𝑟 ℎ → ℎ ≃𝑔𝑟 𝑔))
3		grictr 47830	. . 3 ⊢ ((𝑔 ≃𝑔𝑟 ℎ ∧ ℎ ≃𝑔𝑟 𝑘) → 𝑔 ≃𝑔𝑟 𝑘)
4	3	a1i 11	. 2 ⊢ (𝑔 ∈ UHGraph → ((𝑔 ≃𝑔𝑟 ℎ ∧ ℎ ≃𝑔𝑟 𝑘) → 𝑔 ≃𝑔𝑟 𝑘))
5	1, 2, 4	brinxper 8773	1 ⊢ ( ≃𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2106   ∩ cin 3962   class class class wbr 5148   × cxp 5687   Er wer 8741  UHGraphcuhgr 29088   ≃_𝑔𝑟 cgric 47800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-1o 8505  df-er 8744  df-map 8867  df-uhgr 29090  df-grim 47802  df-gric 47805

This theorem is referenced by: (None)