Theorem gricer 48400

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 48396	. 2 ⊢ (𝑔 ∈ UHGraph → 𝑔 ≃𝑔𝑟 𝑔)
2		gricsym 48397	. 2 ⊢ (𝑔 ∈ UHGraph → (𝑔 ≃𝑔𝑟 ℎ → ℎ ≃𝑔𝑟 𝑔))
3		grictr 48399	. . 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 8673	1 ⊢ ( ≃𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114   ∩ cin 3888   class class class wbr 5085   × cxp 5629   Er wer 8640  UHGraphcuhgr 29125   ≃_𝑔𝑟 cgric 48352

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-1o 8405  df-er 8643  df-map 8775  df-uhgr 29127  df-grim 48354  df-gric 48357

This theorem is referenced by: (None)