Theorem grimf1o 48669

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:𝑉–1-1-onto→𝑊)

Proof of Theorem grimf1o

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grimprop.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		grimprop.w	. . 3 ⊢ 𝑊 = (Vtx‘𝐻)
3		eqid 2763	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4		eqid 2763	. . 3 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
5	1, 2, 3, 4	grimprop 48668	. 2 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))))
6	5	simpld 499	1 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:𝑉–1-1-onto→𝑊)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∀wral 3079  dom cdm 5661   “ cima 5664  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  Vtxcvtx 29346  iEdgciedg 29347   GraphIso cgrim 48660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8822  df-grim 48663

This theorem is referenced by:  uhgrimedg  48676  uhgrimprop  48677  upgrimwlklem4  48685  upgrimwlklem5  48686  upgrimtrlslem2  48690  upgrimpthslem1  48692  upgrimpthslem2  48693  upgrimspths  48695  gricen  48710  clnbgrgrimlem  48718  clnbgrgrim  48719  grimgrtri  48734  uhgrimgrlim  48772