Theorem gricbri 48537

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 48535	. . 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 48536	. . 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)))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∀wral 3079  Vcvv 3457   class class class wbr 5104  dom cdm 5651   “ cima 5654  –1-1-onto→wf1o 6524  ‘cfv 6525  Vtxcvtx 29251  iEdgciedg 29252   ≃_𝑔𝑟 cgric 48497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  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 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-1o 8441  df-map 8814  df-grim 48499  df-gric 48502

This theorem is referenced by: (None)