Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  grictr Structured version   Visualization version   GIF version

Theorem grictr 47892
Description: Graph isomorphism is transitive. (Contributed by AV, 5-Dec-2022.) (Revised by AV, 3-May-2025.)
Assertion
Ref Expression
grictr ((𝑅𝑔𝑟 𝑆𝑆𝑔𝑟 𝑇) → 𝑅𝑔𝑟 𝑇)

Proof of Theorem grictr
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brgric 47881 . 2 (𝑅𝑔𝑟 𝑆 ↔ (𝑅 GraphIso 𝑆) ≠ ∅)
2 brgric 47881 . 2 (𝑆𝑔𝑟 𝑇 ↔ (𝑆 GraphIso 𝑇) ≠ ∅)
3 n0 4353 . . 3 ((𝑅 GraphIso 𝑆) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝑅 GraphIso 𝑆))
4 n0 4353 . . 3 ((𝑆 GraphIso 𝑇) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑆 GraphIso 𝑇))
5 exdistrv 1955 . . . 4 (∃𝑔𝑓(𝑔 ∈ (𝑅 GraphIso 𝑆) ∧ 𝑓 ∈ (𝑆 GraphIso 𝑇)) ↔ (∃𝑔 𝑔 ∈ (𝑅 GraphIso 𝑆) ∧ ∃𝑓 𝑓 ∈ (𝑆 GraphIso 𝑇)))
6 grimco 47880 . . . . . . 7 ((𝑓 ∈ (𝑆 GraphIso 𝑇) ∧ 𝑔 ∈ (𝑅 GraphIso 𝑆)) → (𝑓𝑔) ∈ (𝑅 GraphIso 𝑇))
76ancoms 458 . . . . . 6 ((𝑔 ∈ (𝑅 GraphIso 𝑆) ∧ 𝑓 ∈ (𝑆 GraphIso 𝑇)) → (𝑓𝑔) ∈ (𝑅 GraphIso 𝑇))
8 brgrici 47882 . . . . . 6 ((𝑓𝑔) ∈ (𝑅 GraphIso 𝑇) → 𝑅𝑔𝑟 𝑇)
97, 8syl 17 . . . . 5 ((𝑔 ∈ (𝑅 GraphIso 𝑆) ∧ 𝑓 ∈ (𝑆 GraphIso 𝑇)) → 𝑅𝑔𝑟 𝑇)
109exlimivv 1932 . . . 4 (∃𝑔𝑓(𝑔 ∈ (𝑅 GraphIso 𝑆) ∧ 𝑓 ∈ (𝑆 GraphIso 𝑇)) → 𝑅𝑔𝑟 𝑇)
115, 10sylbir 235 . . 3 ((∃𝑔 𝑔 ∈ (𝑅 GraphIso 𝑆) ∧ ∃𝑓 𝑓 ∈ (𝑆 GraphIso 𝑇)) → 𝑅𝑔𝑟 𝑇)
123, 4, 11syl2anb 598 . 2 (((𝑅 GraphIso 𝑆) ≠ ∅ ∧ (𝑆 GraphIso 𝑇) ≠ ∅) → 𝑅𝑔𝑟 𝑇)
131, 2, 12syl2anb 598 1 ((𝑅𝑔𝑟 𝑆𝑆𝑔𝑟 𝑇) → 𝑅𝑔𝑟 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1779  wcel 2108  wne 2940  c0 4333   class class class wbr 5143  ccom 5689  (class class class)co 7431   GraphIso cgrim 47861  𝑔𝑟 cgric 47862
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-1o 8506  df-map 8868  df-grim 47864  df-gric 47867
This theorem is referenced by:  gricer  47893  grlictr  47975  clnbgr3stgrgrlic  47979
  Copyright terms: Public domain W3C validator