Theorem grimf1o 48538

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 2769	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4		eqid 2769	. . 3 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
5	1, 2, 3, 4	grimprop 48537	. 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 1567  ∃wex 1806   ∈ wcel 2149  ∀wral 3085  dom cdm 5662   “ cima 5665  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411  Vtxcvtx 29287  iEdgciedg 29288   GraphIso cgrim 48529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8826  df-grim 48532

This theorem is referenced by:  uhgrimedg  48545  uhgrimprop  48546  upgrimwlklem4  48554  upgrimwlklem5  48555  upgrimtrlslem2  48559  upgrimpthslem1  48561  upgrimpthslem2  48562  upgrimspths  48564  gricen  48579  clnbgrgrimlem  48587  clnbgrgrim  48588  grimgrtri  48603  uhgrimgrlim  48641