Theorem grimf1o 47872

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 47871	. 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 5623   “ cima 5626  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7353  Vtxcvtx 28959  iEdgciedg 28960   GraphIso cgrim 47863

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 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-map 8762  df-grim 47866

This theorem is referenced by:  uhgrimedg  47879  uhgrimprop  47880  upgrimwlklem4  47888  upgrimwlklem5  47889  upgrimtrlslem2  47893  upgrimpthslem1  47895  upgrimpthslem2  47896  upgrimspths  47898  gricen  47913  clnbgrgrimlem  47921  clnbgrgrim  47922  grimgrtri  47937  uhgrimgrlim  47975