Theorem gricsymb 47952

Description: Graph isomorphism is symmetric in both directions for hypergraphs. (Contributed by AV, 11-Nov-2022.) (Proof shortened by AV, 3-May-2025.)

Assertion

Ref	Expression
gricsymb	⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph) → (𝐴 ≃𝑔𝑟 𝐵 ↔ 𝐵 ≃𝑔𝑟 𝐴))

Proof of Theorem gricsymb

Step	Hyp	Ref	Expression
1		gricsym 47951	. 2 ⊢ (𝐴 ∈ UHGraph → (𝐴 ≃𝑔𝑟 𝐵 → 𝐵 ≃𝑔𝑟 𝐴))
2		gricsym 47951	. 2 ⊢ (𝐵 ∈ UHGraph → (𝐵 ≃𝑔𝑟 𝐴 → 𝐴 ≃𝑔𝑟 𝐵))
3	1, 2	anbiim 641	1 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph) → (𝐴 ≃𝑔𝑟 𝐵 ↔ 𝐵 ≃𝑔𝑟 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2111   class class class wbr 5091  UHGraphcuhgr 29032   ≃_𝑔𝑟 cgric 47906

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-1o 8385  df-map 8752  df-uhgr 29034  df-grim 47908  df-gric 47911

This theorem is referenced by: (None)