Theorem grimfn 48567

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 48566	. 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 6821	. . . . 5 ⊢ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) → 𝑓:(Vtx‘𝑔)⟶(Vtx‘ℎ))
3		fvex 6895	. . . . . 6 ⊢ (Vtx‘ℎ) ∈ V
4		fvex 6895	. . . . . 6 ⊢ (Vtx‘𝑔) ∈ V
5	3, 4	elmap 8869	. . . . 5 ⊢ (𝑓 ∈ ((Vtx‘ℎ) ↑m (Vtx‘𝑔)) ↔ 𝑓:(Vtx‘𝑔)⟶(Vtx‘ℎ))
6	2, 5	sylibr 237	. . . 4 ⊢ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) → 𝑓 ∈ ((Vtx‘ℎ) ↑m (Vtx‘𝑔)))
7	6	adantr 485	. . 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 7444	. . 3 ⊢ ((Vtx‘ℎ) ↑m (Vtx‘𝑔)) ∈ V
9	7, 8	abex 5297	. 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 8067	1 ⊢ GraphIso Fn (V × V)

Syntax hints:   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747  ∀wral 3085  Vcvv 3463  [wsbc 3753   × cxp 5660  dom cdm 5662   “ cima 5665   Fn wfn 6532  ⟶wf 6533  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411   ↑_m cmap 8824  Vtxcvtx 29287  iEdgciedg 29288   GraphIso cgrim 48563

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-csb 3862  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-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  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-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-map 8826  df-grim 48566

This theorem is referenced by:  brgric  48600  gricrel  48607