Theorem gricsymb 47922

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 47921	. 2 ⊢ (𝐴 ∈ UHGraph → (𝐴 ≃𝑔𝑟 𝐵 → 𝐵 ≃𝑔𝑟 𝐴))
2		gricsym 47921	. 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 2109   class class class wbr 5107  UHGraphcuhgr 28983   ≃_𝑔𝑟 cgric 47876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-1o 8434  df-map 8801  df-uhgr 28985  df-grim 47878  df-gric 47881

This theorem is referenced by: (None)