Theorem gricer 48112

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 48108	. 2 ⊢ (𝑔 ∈ UHGraph → 𝑔 ≃𝑔𝑟 𝑔)
2		gricsym 48109	. 2 ⊢ (𝑔 ∈ UHGraph → (𝑔 ≃𝑔𝑟 ℎ → ℎ ≃𝑔𝑟 𝑔))
3		grictr 48111	. . 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 8662	1 ⊢ ( ≃𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2113   ∩ cin 3898   class class class wbr 5096   × cxp 5620   Er wer 8630  UHGraphcuhgr 29078   ≃_𝑔𝑟 cgric 48064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-1o 8395  df-er 8633  df-map 8763  df-uhgr 29080  df-grim 48066  df-gric 48069

This theorem is referenced by: (None)