Theorem gricgrlic 48010

Description: Isomorphic hypergraphs are locally isomorphic. (Contributed by AV, 12-Jun-2025.) (Proof shortened by AV, 11-Jul-2025.)

Assertion

Ref	Expression
gricgrlic	⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝐺 ≃𝑔𝑟 𝐻 → 𝐺 ≃𝑙𝑔𝑟 𝐻))

Proof of Theorem gricgrlic

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brgric 47912	. . 3 ⊢ (𝐺 ≃𝑔𝑟 𝐻 ↔ (𝐺 GraphIso 𝐻) ≠ ∅)
2		n0 4316	. . . 4 ⊢ ((𝐺 GraphIso 𝐻) ≠ ∅ ↔ ∃𝑖 𝑖 ∈ (𝐺 GraphIso 𝐻))
3		uhgrimgrlim 47986	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝑖 ∈ (𝐺 GraphIso 𝐻)) → 𝑖 ∈ (𝐺 GraphLocIso 𝐻))
4		brgrilci 47997	. . . . . . . 8 ⊢ (𝑖 ∈ (𝐺 GraphLocIso 𝐻) → 𝐺 ≃𝑙𝑔𝑟 𝐻)
5	3, 4	syl 17	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝑖 ∈ (𝐺 GraphIso 𝐻)) → 𝐺 ≃𝑙𝑔𝑟 𝐻)
6	5	3expa 1118	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝑖 ∈ (𝐺 GraphIso 𝐻)) → 𝐺 ≃𝑙𝑔𝑟 𝐻)
7	6	expcom 413	. . . . 5 ⊢ (𝑖 ∈ (𝐺 GraphIso 𝐻) → ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → 𝐺 ≃𝑙𝑔𝑟 𝐻))
8	7	exlimiv 1930	. . . 4 ⊢ (∃𝑖 𝑖 ∈ (𝐺 GraphIso 𝐻) → ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → 𝐺 ≃𝑙𝑔𝑟 𝐻))
9	2, 8	sylbi 217	. . 3 ⊢ ((𝐺 GraphIso 𝐻) ≠ ∅ → ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → 𝐺 ≃𝑙𝑔𝑟 𝐻))
10	1, 9	sylbi 217	. 2 ⊢ (𝐺 ≃𝑔𝑟 𝐻 → ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → 𝐺 ≃𝑙𝑔𝑟 𝐻))
11	10	com12 32	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝐺 ≃𝑔𝑟 𝐻 → 𝐺 ≃𝑙𝑔𝑟 𝐻))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∅c0 4296   class class class wbr 5107  (class class class)co 7387  UHGraphcuhgr 28983   GraphIso cgrim 47875   ≃_𝑔𝑟 cgric 47876   GraphLocIso cgrlim 47975   ≃_𝑙𝑔𝑟 cgrlic 47976

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-rep 5234  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-reu 3355  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-vtx 28925  df-iedg 28926  df-edg 28975  df-uhgr 28985  df-clnbgr 47820  df-isubgr 47861  df-grim 47878  df-gric 47881  df-grlim 47977  df-grlic 47980

This theorem is referenced by: (None)