Theorem grimfn 48067

Description: The graph isomorphism function is a well-defined function. (Contributed by AV, 28-Apr-2025.)

Assertion

Ref	Expression
grimfn	⊢ GraphIso Fn (V × V)

Proof of Theorem grimfn

Dummy variables 𝑒 𝑑 𝑓 𝑔 ℎ 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-grim 48066	. 2 ⊢ GraphIso = (𝑔 ∈ V, ℎ ∈ V ↦ {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ∧ ∃𝑗[(iEdg‘𝑔) / 𝑒][(iEdg‘ℎ) / 𝑑](𝑗:dom 𝑒–1-1-onto→dom 𝑑 ∧ ∀𝑖 ∈ dom 𝑒(𝑑‘(𝑗‘𝑖)) = (𝑓 “ (𝑒‘𝑖))))})
2		f1of 6772	. . . . 5 ⊢ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) → 𝑓:(Vtx‘𝑔)⟶(Vtx‘ℎ))
3		fvex 6845	. . . . . 6 ⊢ (Vtx‘ℎ) ∈ V
4		fvex 6845	. . . . . 6 ⊢ (Vtx‘𝑔) ∈ V
5	3, 4	elmap 8807	. . . . 5 ⊢ (𝑓 ∈ ((Vtx‘ℎ) ↑m (Vtx‘𝑔)) ↔ 𝑓:(Vtx‘𝑔)⟶(Vtx‘ℎ))
6	2, 5	sylibr 234	. . . 4 ⊢ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) → 𝑓 ∈ ((Vtx‘ℎ) ↑m (Vtx‘𝑔)))
7	6	adantr 480	. . 3 ⊢ ((𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ∧ ∃𝑗[(iEdg‘𝑔) / 𝑒][(iEdg‘ℎ) / 𝑑](𝑗:dom 𝑒–1-1-onto→dom 𝑑 ∧ ∀𝑖 ∈ dom 𝑒(𝑑‘(𝑗‘𝑖)) = (𝑓 “ (𝑒‘𝑖)))) → 𝑓 ∈ ((Vtx‘ℎ) ↑m (Vtx‘𝑔)))
8		ovex 7389	. . 3 ⊢ ((Vtx‘ℎ) ↑m (Vtx‘𝑔)) ∈ V
9	7, 8	abex 5269	. 2 ⊢ {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ∧ ∃𝑗[(iEdg‘𝑔) / 𝑒][(iEdg‘ℎ) / 𝑑](𝑗:dom 𝑒–1-1-onto→dom 𝑑 ∧ ∀𝑖 ∈ dom 𝑒(𝑑‘(𝑗‘𝑖)) = (𝑓 “ (𝑒‘𝑖))))} ∈ V
10	1, 9	fnmpoi 8012	1 ⊢ GraphIso Fn (V × V)

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2712  ∀wral 3049  Vcvv 3438  [wsbc 3738   × cxp 5620  dom cdm 5622   “ cima 5625   Fn wfn 6485  ⟶wf 6486  –1-1-onto→wf1o 6489  ‘cfv 6490  (class class class)co 7356   ↑_m cmap 8761  Vtxcvtx 29018  iEdgciedg 29019   GraphIso cgrim 48063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8763  df-grim 48066

This theorem is referenced by:  brgric  48100  gricrel  48107