Theorem dfgric2 47533

Description: Alternate, explicit definition of the "is isomorphic to" relation for two graphs. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 5-May-2025.)

Hypotheses

Ref	Expression
dfgric2.v	⊢ 𝑉 = (Vtx‘𝐴)
dfgric2.w	⊢ 𝑊 = (Vtx‘𝐵)
dfgric2.i	⊢ 𝐼 = (iEdg‘𝐴)
dfgric2.j	⊢ 𝐽 = (iEdg‘𝐵)

Assertion

Ref	Expression
dfgric2	⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴 ≃𝑔𝑟 𝐵 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))

Distinct variable groups:   𝐴,𝑓,𝑔,𝑖   𝐵,𝑓,𝑔,𝑖   𝑖,𝐼   𝑓,𝑋   𝑓,𝑌

Allowed substitution hints:   𝐼(𝑓,𝑔)   𝐽(𝑓,𝑔,𝑖)   𝑉(𝑓,𝑔,𝑖)   𝑊(𝑓,𝑔,𝑖)   𝑋(𝑔,𝑖)   𝑌(𝑔,𝑖)

Proof of Theorem dfgric2

Step	Hyp	Ref	Expression
1		brgric 47530	. . 3 ⊢ (𝐴 ≃𝑔𝑟 𝐵 ↔ (𝐴 GraphIso 𝐵) ≠ ∅)
2		n0 4359	. . 3 ⊢ ((𝐴 GraphIso 𝐵) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐴 GraphIso 𝐵))
3	1, 2	bitri 274	. 2 ⊢ (𝐴 ≃𝑔𝑟 𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴 GraphIso 𝐵))
4		vex 3476	. . . 4 ⊢ 𝑓 ∈ V
5		dfgric2.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐴)
6		dfgric2.w	. . . . . 6 ⊢ 𝑊 = (Vtx‘𝐵)
7		dfgric2.i	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐴)
8		dfgric2.j	. . . . . 6 ⊢ 𝐽 = (iEdg‘𝐵)
9	5, 6, 7, 8	isgrim 47517	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑓 ∈ V) → (𝑓 ∈ (𝐴 GraphIso 𝐵) ↔ (𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝐽‘(𝑔‘𝑖)) = (𝑓 “ (𝐼‘𝑖))))))
10		eqcom 2736	. . . . . . . . 9 ⊢ ((𝐽‘(𝑔‘𝑖)) = (𝑓 “ (𝐼‘𝑖)) ↔ (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))
11	10	ralbii 3086	. . . . . . . 8 ⊢ (∀𝑖 ∈ dom 𝐼(𝐽‘(𝑔‘𝑖)) = (𝑓 “ (𝐼‘𝑖)) ↔ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))
12	11	anbi2i 621	. . . . . . 7 ⊢ ((𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝐽‘(𝑔‘𝑖)) = (𝑓 “ (𝐼‘𝑖))) ↔ (𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))
13	12	exbii 1843	. . . . . 6 ⊢ (∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝐽‘(𝑔‘𝑖)) = (𝑓 “ (𝐼‘𝑖))) ↔ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))
14	13	anbi2i 621	. . . . 5 ⊢ ((𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝐽‘(𝑔‘𝑖)) = (𝑓 “ (𝐼‘𝑖)))) ↔ (𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))
15	9, 14	bitrdi 286	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑓 ∈ V) → (𝑓 ∈ (𝐴 GraphIso 𝐵) ↔ (𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
16	4, 15	mp3an3 1447	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝑓 ∈ (𝐴 GraphIso 𝐵) ↔ (𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
17	16	exbidv 1917	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (∃𝑓 𝑓 ∈ (𝐴 GraphIso 𝐵) ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
18	3, 17	bitrid 282	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴 ≃𝑔𝑟 𝐵 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1534  ∃wex 1774   ∈ wcel 2100   ≠ wne 2933  ∀wral 3054  Vcvv 3472  ∅c0 4335   class class class wbr 5156  dom cdm 5686   “ cima 5689  –1-1-onto→wf1o 6557  ‘cfv 6558  (class class class)co 7430  Vtxcvtx 28955  iEdgciedg 28956   GraphIso cgrim 47510   ≃_𝑔𝑟 cgric 47511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-sep 5307  ax-nul 5314  ax-pow 5373  ax-pr 5437  ax-un 7751

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-sbc 3789  df-csb 3905  df-dif 3962  df-un 3964  df-in 3966  df-ss 3976  df-nul 4336  df-if 4537  df-pw 4612  df-sn 4637  df-pr 4639  df-op 4643  df-uni 4919  df-iun 5008  df-br 5157  df-opab 5219  df-mpt 5240  df-id 5584  df-xp 5692  df-rel 5693  df-cnv 5694  df-co 5695  df-dm 5696  df-rn 5697  df-res 5698  df-ima 5699  df-suc 6386  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8011  df-2nd 8012  df-1o 8504  df-map 8865  df-grim 47513  df-gric 47516

This theorem is referenced by:  gricbri  47534  gricushgr  47535  ushggricedg  47545  isubgrgrim  47546