Theorem gricbri 48983

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 48981	. . 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 48982	. . 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 18	. 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 270	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)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3077  Vcvv 3451   class class class wbr 5103  dom cdm 5651   “ cima 5654  –1-1-onto→wf1o 6536  ‘cfv 6537  Vtxcvtx 29567  iEdgciedg 29568   ≃_𝑔𝑟 cgric 48943

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-1o 8469  df-map 8842  df-grim 48945  df-gric 48948

This theorem is used by: (None)