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Theorem dfgric2 47884
Description: Alternate, explicit definition of the "is isomorphic to" relation for two graphs. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 5-May-2025.)
Hypotheses
Ref Expression
dfgric2.v 𝑉 = (Vtx‘𝐴)
dfgric2.w 𝑊 = (Vtx‘𝐵)
dfgric2.i 𝐼 = (iEdg‘𝐴)
dfgric2.j 𝐽 = (iEdg‘𝐵)
Assertion
Ref Expression
dfgric2 ((𝐴𝑋𝐵𝑌) → (𝐴𝑔𝑟 𝐵 ↔ ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∃𝑔(𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))
Distinct variable groups:   𝐴,𝑓,𝑔,𝑖   𝐵,𝑓,𝑔,𝑖   𝑖,𝐼   𝑓,𝑋   𝑓,𝑌
Allowed substitution hints:   𝐼(𝑓,𝑔)   𝐽(𝑓,𝑔,𝑖)   𝑉(𝑓,𝑔,𝑖)   𝑊(𝑓,𝑔,𝑖)   𝑋(𝑔,𝑖)   𝑌(𝑔,𝑖)

Proof of Theorem dfgric2
StepHypRef Expression
1 brgric 47881 . . 3 (𝐴𝑔𝑟 𝐵 ↔ (𝐴 GraphIso 𝐵) ≠ ∅)
2 n0 4353 . . 3 ((𝐴 GraphIso 𝐵) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐴 GraphIso 𝐵))
31, 2bitri 275 . 2 (𝐴𝑔𝑟 𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴 GraphIso 𝐵))
4 vex 3484 . . . 4 𝑓 ∈ V
5 dfgric2.v . . . . . 6 𝑉 = (Vtx‘𝐴)
6 dfgric2.w . . . . . 6 𝑊 = (Vtx‘𝐵)
7 dfgric2.i . . . . . 6 𝐼 = (iEdg‘𝐴)
8 dfgric2.j . . . . . 6 𝐽 = (iEdg‘𝐵)
95, 6, 7, 8isgrim 47868 . . . . 5 ((𝐴𝑋𝐵𝑌𝑓 ∈ V) → (𝑓 ∈ (𝐴 GraphIso 𝐵) ↔ (𝑓:𝑉1-1-onto𝑊 ∧ ∃𝑔(𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝐽‘(𝑔𝑖)) = (𝑓 “ (𝐼𝑖))))))
10 eqcom 2744 . . . . . . . . 9 ((𝐽‘(𝑔𝑖)) = (𝑓 “ (𝐼𝑖)) ↔ (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))
1110ralbii 3093 . . . . . . . 8 (∀𝑖 ∈ dom 𝐼(𝐽‘(𝑔𝑖)) = (𝑓 “ (𝐼𝑖)) ↔ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))
1211anbi2i 623 . . . . . . 7 ((𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝐽‘(𝑔𝑖)) = (𝑓 “ (𝐼𝑖))) ↔ (𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))
1312exbii 1848 . . . . . 6 (∃𝑔(𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝐽‘(𝑔𝑖)) = (𝑓 “ (𝐼𝑖))) ↔ ∃𝑔(𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))
1413anbi2i 623 . . . . 5 ((𝑓:𝑉1-1-onto𝑊 ∧ ∃𝑔(𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝐽‘(𝑔𝑖)) = (𝑓 “ (𝐼𝑖)))) ↔ (𝑓:𝑉1-1-onto𝑊 ∧ ∃𝑔(𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))))
159, 14bitrdi 287 . . . 4 ((𝐴𝑋𝐵𝑌𝑓 ∈ V) → (𝑓 ∈ (𝐴 GraphIso 𝐵) ↔ (𝑓:𝑉1-1-onto𝑊 ∧ ∃𝑔(𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))
164, 15mp3an3 1452 . . 3 ((𝐴𝑋𝐵𝑌) → (𝑓 ∈ (𝐴 GraphIso 𝐵) ↔ (𝑓:𝑉1-1-onto𝑊 ∧ ∃𝑔(𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))
1716exbidv 1921 . 2 ((𝐴𝑋𝐵𝑌) → (∃𝑓 𝑓 ∈ (𝐴 GraphIso 𝐵) ↔ ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∃𝑔(𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))
183, 17bitrid 283 1 ((𝐴𝑋𝐵𝑌) → (𝐴𝑔𝑟 𝐵 ↔ ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∃𝑔(𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  Vcvv 3480  c0 4333   class class class wbr 5143  dom cdm 5685  cima 5688  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  Vtxcvtx 29013  iEdgciedg 29014   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:  gricbri  47885  gricushgr  47886  ushggricedg  47896  isubgrgrim  47897
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