Theorem grimfn 48502

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 48501	. 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 6807	. . . . 5 ⊢ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) → 𝑓:(Vtx‘𝑔)⟶(Vtx‘ℎ))
3		fvex 6881	. . . . . 6 ⊢ (Vtx‘ℎ) ∈ V
4		fvex 6881	. . . . . 6 ⊢ (Vtx‘𝑔) ∈ V
5	3, 4	elmap 8854	. . . . 5 ⊢ (𝑓 ∈ ((Vtx‘ℎ) ↑m (Vtx‘𝑔)) ↔ 𝑓:(Vtx‘𝑔)⟶(Vtx‘ℎ))
6	2, 5	sylibr 236	. . . 4 ⊢ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) → 𝑓 ∈ ((Vtx‘ℎ) ↑m (Vtx‘𝑔)))
7	6	adantr 484	. . 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 7430	. . 3 ⊢ ((Vtx‘ℎ) ↑m (Vtx‘𝑔)) ∈ V
9	7, 8	abex 5283	. 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 8052	1 ⊢ GraphIso Fn (V × V)

Syntax hints:   ∧ wa 399   = wceq 1561  ∃wex 1800   ∈ wcel 2143  {cab 2741  ∀wral 3077  Vcvv 3455  [wsbc 3745   × cxp 5646  dom cdm 5648   “ cima 5651   Fn wfn 6517  ⟶wf 6518  –1-1-onto→wf1o 6521  ‘cfv 6522  (class class class)co 7397   ↑_m cmap 8809  Vtxcvtx 29198  iEdgciedg 29199   GraphIso cgrim 48498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-f1o 6529  df-fv 6530  df-ov 7400  df-oprab 7401  df-mpo 7402  df-1st 7971  df-2nd 7972  df-map 8811  df-grim 48501

This theorem is referenced by:  brgric  48535  gricrel  48542