Theorem dfgric2 48929

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 48926	. . 3 ⊢ (𝐴 ≃𝑔𝑟 𝐵 ↔ (𝐴 GraphIso 𝐵) ≠ ∅)
2		n0 4300	. . 3 ⊢ ((𝐴 GraphIso 𝐵) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐴 GraphIso 𝐵))
3	1, 2	bitri 278	. 2 ⊢ (𝐴 ≃𝑔𝑟 𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴 GraphIso 𝐵))
4		vex 3454	. . . 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 48896	. . . . 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 2767	. . . . . . . . 9 ⊢ ((𝐽‘(𝑔‘𝑖)) = (𝑓 “ (𝐼‘𝑖)) ↔ (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))
11	10	ralbii 3108	. . . . . . . 8 ⊢ (∀𝑖 ∈ dom 𝐼(𝐽‘(𝑔‘𝑖)) = (𝑓 “ (𝐼‘𝑖)) ↔ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))
12	11	anbi2i 635	. . . . . . 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 1881	. . . . . 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 635	. . . . 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 290	. . . 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 1479	. . 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 1954	. 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 286	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))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  Vcvv 3450  ∅c0 4279   class class class wbr 5103  dom cdm 5648   “ cima 5651  –1-1-onto→wf1o 6527  ‘cfv 6528  (class class class)co 7409  Vtxcvtx 29493  iEdgciedg 29494   GraphIso cgrim 48889   ≃_𝑔𝑟 cgric 48890

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 2732  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  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 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7985  df-2nd 7986  df-1o 8455  df-map 8828  df-grim 48892  df-gric 48895

This theorem is used by:  gricbri  48930  gricushgr  48931  ushggricedg  48941  isubgrgrim  48943