Theorem dfgric2 48228

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 48225	. . 3 ⊢ (𝐴 ≃𝑔𝑟 𝐵 ↔ (𝐴 GraphIso 𝐵) ≠ ∅)
2		n0 4306	. . 3 ⊢ ((𝐴 GraphIso 𝐵) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐴 GraphIso 𝐵))
3	1, 2	bitri 275	. 2 ⊢ (𝐴 ≃𝑔𝑟 𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴 GraphIso 𝐵))
4		vex 3445	. . . 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 48195	. . . . 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 2744	. . . . . . . . 9 ⊢ ((𝐽‘(𝑔‘𝑖)) = (𝑓 “ (𝐼‘𝑖)) ↔ (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))
11	10	ralbii 3083	. . . . . . . 8 ⊢ (∀𝑖 ∈ dom 𝐼(𝐽‘(𝑔‘𝑖)) = (𝑓 “ (𝐼‘𝑖)) ↔ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))
12	11	anbi2i 624	. . . . . . 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 1850	. . . . . 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 624	. . . . 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 287	. . . 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 1453	. . 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 1923	. 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 283	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3441  ∅c0 4286   class class class wbr 5099  dom cdm 5625   “ cima 5628  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7360  Vtxcvtx 29073  iEdgciedg 29074   GraphIso cgrim 48188   ≃_𝑔𝑟 cgric 48189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-1o 8399  df-map 8769  df-grim 48191  df-gric 48194

This theorem is referenced by:  gricbri  48229  gricushgr  48230  ushggricedg  48240  isubgrgrim  48242