Theorem isomgr 46005

Description: The isomorphy relation for two graphs. (Contributed by AV, 11-Nov-2022.)

Hypotheses

Ref	Expression
isomgr.v	⊢ 𝑉 = (Vtx‘𝐴)
isomgr.w	⊢ 𝑊 = (Vtx‘𝐵)
isomgr.i	⊢ 𝐼 = (iEdg‘𝐴)
isomgr.j	⊢ 𝐽 = (iEdg‘𝐵)

Assertion

Ref	Expression
isomgr	⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴 IsomGr 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)   𝑋(𝑓,𝑔,𝑖)   𝑌(𝑓,𝑔,𝑖)

Proof of Theorem isomgr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2737	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑓 = 𝑓)
2		fveq2 6842	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (Vtx‘𝑥) = (Vtx‘𝐴))
3	2	adantr 481	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (Vtx‘𝑥) = (Vtx‘𝐴))
4		isomgr.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐴)
5	3, 4	eqtr4di 2794	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (Vtx‘𝑥) = 𝑉)
6		fveq2 6842	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (Vtx‘𝑦) = (Vtx‘𝐵))
7	6	adantl 482	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (Vtx‘𝑦) = (Vtx‘𝐵))
8		isomgr.w	. . . . . 6 ⊢ 𝑊 = (Vtx‘𝐵)
9	7, 8	eqtr4di 2794	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (Vtx‘𝑦) = 𝑊)
10	1, 5, 9	f1oeq123d 6778	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑓:(Vtx‘𝑥)–1-1-onto→(Vtx‘𝑦) ↔ 𝑓:𝑉–1-1-onto→𝑊))
11		eqidd 2737	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑔 = 𝑔)
12		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (iEdg‘𝑥) = (iEdg‘𝐴))
13	12	adantr 481	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (iEdg‘𝑥) = (iEdg‘𝐴))
14		isomgr.i	. . . . . . . . 9 ⊢ 𝐼 = (iEdg‘𝐴)
15	13, 14	eqtr4di 2794	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (iEdg‘𝑥) = 𝐼)
16	15	dmeqd 5861	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → dom (iEdg‘𝑥) = dom 𝐼)
17		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (iEdg‘𝑦) = (iEdg‘𝐵))
18	17	adantl 482	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (iEdg‘𝑦) = (iEdg‘𝐵))
19		isomgr.j	. . . . . . . . 9 ⊢ 𝐽 = (iEdg‘𝐵)
20	18, 19	eqtr4di 2794	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (iEdg‘𝑦) = 𝐽)
21	20	dmeqd 5861	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → dom (iEdg‘𝑦) = dom 𝐽)
22	11, 16, 21	f1oeq123d 6778	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ↔ 𝑔:dom 𝐼–1-1-onto→dom 𝐽))
23	15	fveq1d 6844	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((iEdg‘𝑥)‘𝑖) = (𝐼‘𝑖))
24	23	imaeq2d 6013	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑓 “ ((iEdg‘𝑥)‘𝑖)) = (𝑓 “ (𝐼‘𝑖)))
25	20	fveq1d 6844	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((iEdg‘𝑦)‘(𝑔‘𝑖)) = (𝐽‘(𝑔‘𝑖)))
26	24, 25	eqeq12d 2752	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔‘𝑖)) ↔ (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))
27	16, 26	raleqbidv 3319	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔‘𝑖)) ↔ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))
28	22, 27	anbi12d 631	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ∧ ∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔‘𝑖))) ↔ (𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))
29	28	exbidv 1924	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑔(𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ∧ ∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔‘𝑖))) ↔ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))
30	10, 29	anbi12d 631	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑓:(Vtx‘𝑥)–1-1-onto→(Vtx‘𝑦) ∧ ∃𝑔(𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ∧ ∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔‘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))))))
31	30	exbidv 1924	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑓(𝑓:(Vtx‘𝑥)–1-1-onto→(Vtx‘𝑦) ∧ ∃𝑔(𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ∧ ∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔‘𝑖)))) ↔ ∃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))))))
32		df-isomgr 46003	. 2 ⊢ IsomGr = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓(𝑓:(Vtx‘𝑥)–1-1-onto→(Vtx‘𝑦) ∧ ∃𝑔(𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ∧ ∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔‘𝑖))))}
33	31, 32	brabga 5491	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴 IsomGr 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 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3064   class class class wbr 5105  dom cdm 5633   “ cima 5636  –1-1-onto→wf1o 6495  ‘cfv 6496  Vtxcvtx 27947  iEdgciedg 27948   IsomGr cisomgr 46001

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isomgr 46003

This theorem is referenced by:  isisomgr  46006  isomgreqve  46007  isomushgr  46008  isomgrsym  46018  isomgrtr  46021  ushrisomgr  46023