Theorem grimfn 47852

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 47851	. 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 6782	. . . . 5 ⊢ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) → 𝑓:(Vtx‘𝑔)⟶(Vtx‘ℎ))
3		fvex 6853	. . . . . 6 ⊢ (Vtx‘ℎ) ∈ V
4		fvex 6853	. . . . . 6 ⊢ (Vtx‘𝑔) ∈ V
5	3, 4	elmap 8821	. . . . 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 7402	. . 3 ⊢ ((Vtx‘ℎ) ↑m (Vtx‘𝑔)) ∈ V
9	7, 8	abex 5276	. 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 8028	1 ⊢ GraphIso Fn (V × V)

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  ∀wral 3044  Vcvv 3444  [wsbc 3750   × cxp 5629  dom cdm 5631   “ cima 5634   Fn wfn 6494  ⟶wf 6495  –1-1-onto→wf1o 6498  ‘cfv 6499  (class class class)co 7369   ↑_m cmap 8776  Vtxcvtx 28899  iEdgciedg 28900   GraphIso cgrim 47848

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 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-map 8778  df-grim 47851

This theorem is referenced by:  brgric  47885  gricrel  47892