Theorem grimf1o 48689

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 2766	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4		eqid 2766	. . 3 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
5	1, 2, 3, 4	grimprop 48688	. 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 500	1 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:𝑉–1-1-onto→𝑊)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3082  dom cdm 5666   “ cima 5669  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7423  Vtxcvtx 29383  iEdgciedg 29384   GraphIso cgrim 48680

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-map 8835  df-grim 48683

This theorem is used by:  uhgrimedg  48696  uhgrimprop  48697  upgrimwlklem4  48705  upgrimwlklem5  48706  upgrimtrlslem2  48710  upgrimpthslem1  48712  upgrimpthslem2  48713  upgrimspths  48715  gricen  48730  clnbgrgrimlem  48738  clnbgrgrim  48739  grimgrtri  48754  uhgrimgrlim  48792