Theorem gricsymb 48398

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 48397	. 2 ⊢ (𝐴 ∈ UHGraph → (𝐴 ≃𝑔𝑟 𝐵 → 𝐵 ≃𝑔𝑟 𝐴))
2		gricsym 48397	. 2 ⊢ (𝐵 ∈ UHGraph → (𝐵 ≃𝑔𝑟 𝐴 → 𝐴 ≃𝑔𝑟 𝐵))
3	1, 2	anbiim 642	1 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph) → (𝐴 ≃𝑔𝑟 𝐵 ↔ 𝐵 ≃𝑔𝑟 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114   class class class wbr 5085  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-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-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-map 8775  df-uhgr 29127  df-grim 48354  df-gric 48357

This theorem is referenced by: (None)