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Theorem dfgric2 47822
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 47819 . . 3 (𝐴𝑔𝑟 𝐵 ↔ (𝐴 GraphIso 𝐵) ≠ ∅)
2 n0 4359 . . 3 ((𝐴 GraphIso 𝐵) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐴 GraphIso 𝐵))
31, 2bitri 275 . 2 (𝐴𝑔𝑟 𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴 GraphIso 𝐵))
4 vex 3482 . . . 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 47806 . . . . 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 2742 . . . . . . . . 9 ((𝐽‘(𝑔𝑖)) = (𝑓 “ (𝐼𝑖)) ↔ (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))
1110ralbii 3091 . . . . . . . 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 1845 . . . . . 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 1449 . . 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 1919 . 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 1086   = wceq 1537  wex 1776  wcel 2106  wne 2938  wral 3059  Vcvv 3478  c0 4339   class class class wbr 5148  dom cdm 5689  cima 5692  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  Vtxcvtx 29028  iEdgciedg 29029   GraphIso cgrim 47799  𝑔𝑟 cgric 47800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-1o 8505  df-map 8867  df-grim 47802  df-gric 47805
This theorem is referenced by:  gricbri  47823  gricushgr  47824  ushggricedg  47834  isubgrgrim  47835
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