Theorem gricer 48415

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 48411	. 2 ⊢ (𝑔 ∈ UHGraph → 𝑔 ≃𝑔𝑟 𝑔)
2		gricsym 48412	. 2 ⊢ (𝑔 ∈ UHGraph → (𝑔 ≃𝑔𝑟 ℎ → ℎ ≃𝑔𝑟 𝑔))
3		grictr 48414	. . 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 8663	1 ⊢ ( ≃𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119   ∩ cin 3882   class class class wbr 5072   × cxp 5616   Er wer 8630  UHGraphcuhgr 29143   ≃_𝑔𝑟 cgric 48367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-1o 8395  df-er 8633  df-map 8765  df-uhgr 29145  df-grim 48369  df-gric 48372

This theorem is referenced by: (None)