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Theorem grimf1o 47808
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 2735 . . 3 (iEdg‘𝐺) = (iEdg‘𝐺)
4 eqid 2735 . . 3 (iEdg‘𝐻) = (iEdg‘𝐻)
51, 2, 3, 4grimprop 47807 . 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 1537  wex 1776  wcel 2106  wral 3059  dom cdm 5689  cima 5692  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  Vtxcvtx 29028  iEdgciedg 29029   GraphIso cgrim 47799
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-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-br 5149  df-opab 5211  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-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-map 8867  df-grim 47802
This theorem is referenced by:  gricen  47832  clnbgrgrimlem  47839  clnbgrgrim  47840  grimgrtri  47852  uhgrimgrlim  47890
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