Theorem grimfn 47751

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 47750	. 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 6864	. . . . 5 ⊢ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) → 𝑓:(Vtx‘𝑔)⟶(Vtx‘ℎ))
3		fvex 6935	. . . . . 6 ⊢ (Vtx‘ℎ) ∈ V
4		fvex 6935	. . . . . 6 ⊢ (Vtx‘𝑔) ∈ V
5	3, 4	elmap 8931	. . . . 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 7483	. . 3 ⊢ ((Vtx‘ℎ) ↑m (Vtx‘𝑔)) ∈ V
9	7, 8	abex 5344	. 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 8113	1 ⊢ GraphIso Fn (V × V)

Syntax hints:   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717  ∀wral 3067  Vcvv 3488  [wsbc 3804   × cxp 5698  dom cdm 5700   “ cima 5703   Fn wfn 6570  ⟶wf 6571  –1-1-onto→wf1o 6574  ‘cfv 6575  (class class class)co 7450   ↑_m cmap 8886  Vtxcvtx 29033  iEdgciedg 29034   GraphIso cgrim 47747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-f1o 6582  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-1st 8032  df-2nd 8033  df-map 8888  df-grim 47750

This theorem is referenced by:  brgric  47767  gricrel  47774