Theorem gricbri 47823

Description: Implications of two graphs being isomorphic. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 5-May-2025.) (Proof shortened by AV, 12-Jun-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
gricbri	⊢ (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

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

Proof of Theorem gricbri

Step	Hyp	Ref	Expression
1		gricrcl 47821	. . 3 ⊢ (𝐴 ≃𝑔𝑟 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
2		dfgric2.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐴)
3		dfgric2.w	. . . 4 ⊢ 𝑊 = (Vtx‘𝐵)
4		dfgric2.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐴)
5		dfgric2.j	. . . 4 ⊢ 𝐽 = (iEdg‘𝐵)
6	2, 3, 4, 5	dfgric2 47822	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (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))))))
7	1, 6	syl 17	. 2 ⊢ (𝐴 ≃𝑔𝑟 𝐵 → (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))))))
8	7	ibi 267	1 ⊢ (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 206   ∧ wa 395   = wceq 1537  ∃wex 1776   ∈ wcel 2106  ∀wral 3059  Vcvv 3478   class class class wbr 5148  dom cdm 5689   “ cima 5692  –1-1-onto→wf1o 6562  ‘cfv 6563  Vtxcvtx 29028  iEdgciedg 29029   ≃_𝑔𝑟 cgric 47800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-1o 8505  df-map 8867  df-grim 47802  df-gric 47805

This theorem is referenced by: (None)