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Theorem dfgric2 47768
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 47765 . . 3 (𝐴𝑔𝑟 𝐵 ↔ (𝐴 GraphIso 𝐵) ≠ ∅)
2 n0 4376 . . 3 ((𝐴 GraphIso 𝐵) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐴 GraphIso 𝐵))
31, 2bitri 275 . 2 (𝐴𝑔𝑟 𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴 GraphIso 𝐵))
4 vex 3492 . . . 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 47752 . . . . 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 2747 . . . . . . . . 9 ((𝐽‘(𝑔𝑖)) = (𝑓 “ (𝐼𝑖)) ↔ (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))
1110ralbii 3099 . . . . . . . 8 (∀𝑖 ∈ dom 𝐼(𝐽‘(𝑔𝑖)) = (𝑓 “ (𝐼𝑖)) ↔ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))
1211anbi2i 622 . . . . . . 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 1846 . . . . . 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 622 . . . . 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 1450 . . 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 1920 . 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 1537  wex 1777  wcel 2108  wne 2946  wral 3067  Vcvv 3488  c0 4352   class class class wbr 5166  dom cdm 5700  cima 5703  1-1-ontowf1o 6572  cfv 6573  (class class class)co 7448  Vtxcvtx 29031  iEdgciedg 29032   GraphIso cgrim 47745  𝑔𝑟 cgric 47746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-1o 8522  df-map 8886  df-grim 47748  df-gric 47751
This theorem is referenced by:  gricbri  47769  gricushgr  47770  ushggricedg  47780  isubgrgrim  47781
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