Theorem grimf1o 47881

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 2729	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4		eqid 2729	. . 3 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
5	1, 2, 3, 4	grimprop 47880	. 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 494	1 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:𝑉–1-1-onto→𝑊)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  dom cdm 5638   “ cima 5641  –1-1-onto→wf1o 6510  ‘cfv 6511  (class class class)co 7387  Vtxcvtx 28923  iEdgciedg 28924   GraphIso cgrim 47872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-map 8801  df-grim 47875

This theorem is referenced by:  uhgrimedg  47888  uhgrimprop  47889  upgrimwlklem4  47897  upgrimwlklem5  47898  upgrimtrlslem2  47902  upgrimpthslem1  47904  upgrimpthslem2  47905  upgrimspths  47907  gricen  47922  clnbgrgrimlem  47930  clnbgrgrim  47931  grimgrtri  47945  uhgrimgrlim  47983