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Theorem grimf1o 47875
Description: An isomorphism of graphs is a bijection between their vertices. (Contributed by AV, 29-Apr-2025.)
Hypotheses
Ref Expression
grimprop.v 𝑉 = (Vtx‘𝐺)
grimprop.w 𝑊 = (Vtx‘𝐻)
Assertion
Ref Expression
grimf1o (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:𝑉1-1-onto𝑊)

Proof of Theorem grimf1o
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grimprop.v . . 3 𝑉 = (Vtx‘𝐺)
2 grimprop.w . . 3 𝑊 = (Vtx‘𝐻)
3 eqid 2736 . . 3 (iEdg‘𝐺) = (iEdg‘𝐺)
4 eqid 2736 . . 3 (iEdg‘𝐻) = (iEdg‘𝐻)
51, 2, 3, 4grimprop 47874 . 2 (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹:𝑉1-1-onto𝑊 ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))))
65simpld 494 1 (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:𝑉1-1-onto𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wex 1778  wcel 2107  wral 3060  dom cdm 5684  cima 5687  1-1-ontowf1o 6559  cfv 6560  (class class class)co 7432  Vtxcvtx 29014  iEdgciedg 29015   GraphIso cgrim 47866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-grim 47869
This theorem is referenced by:  gricen  47899  clnbgrgrimlem  47906  clnbgrgrim  47907  grimgrtri  47921  uhgrimgrlim  47959
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