Theorem isomgr 45163

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 2739	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑓 = 𝑓)
2		fveq2 6756	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (Vtx‘𝑥) = (Vtx‘𝐴))
3	2	adantr 480	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (Vtx‘𝑥) = (Vtx‘𝐴))
4		isomgr.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐴)
5	3, 4	eqtr4di 2797	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (Vtx‘𝑥) = 𝑉)
6		fveq2 6756	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (Vtx‘𝑦) = (Vtx‘𝐵))
7	6	adantl 481	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (Vtx‘𝑦) = (Vtx‘𝐵))
8		isomgr.w	. . . . . 6 ⊢ 𝑊 = (Vtx‘𝐵)
9	7, 8	eqtr4di 2797	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (Vtx‘𝑦) = 𝑊)
10	1, 5, 9	f1oeq123d 6694	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑓:(Vtx‘𝑥)–1-1-onto→(Vtx‘𝑦) ↔ 𝑓:𝑉–1-1-onto→𝑊))
11		eqidd 2739	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑔 = 𝑔)
12		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (iEdg‘𝑥) = (iEdg‘𝐴))
13	12	adantr 480	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (iEdg‘𝑥) = (iEdg‘𝐴))
14		isomgr.i	. . . . . . . . 9 ⊢ 𝐼 = (iEdg‘𝐴)
15	13, 14	eqtr4di 2797	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (iEdg‘𝑥) = 𝐼)
16	15	dmeqd 5803	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → dom (iEdg‘𝑥) = dom 𝐼)
17		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (iEdg‘𝑦) = (iEdg‘𝐵))
18	17	adantl 481	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (iEdg‘𝑦) = (iEdg‘𝐵))
19		isomgr.j	. . . . . . . . 9 ⊢ 𝐽 = (iEdg‘𝐵)
20	18, 19	eqtr4di 2797	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (iEdg‘𝑦) = 𝐽)
21	20	dmeqd 5803	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → dom (iEdg‘𝑦) = dom 𝐽)
22	11, 16, 21	f1oeq123d 6694	. . . . . 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 6758	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((iEdg‘𝑥)‘𝑖) = (𝐼‘𝑖))
24	23	imaeq2d 5958	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑓 “ ((iEdg‘𝑥)‘𝑖)) = (𝑓 “ (𝐼‘𝑖)))
25	20	fveq1d 6758	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((iEdg‘𝑦)‘(𝑔‘𝑖)) = (𝐽‘(𝑔‘𝑖)))
26	24, 25	eqeq12d 2754	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔‘𝑖)) ↔ (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))
27	16, 26	raleqbidv 3327	. . . . . 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 630	. . . . 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 1925	. . . 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 630	. . 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 1925	. 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 45161	. 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 5440	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 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∀wral 3063   class class class wbr 5070  dom cdm 5580   “ cima 5583  –1-1-onto→wf1o 6417  ‘cfv 6418  Vtxcvtx 27269  iEdgciedg 27270   IsomGr cisomgr 45159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isomgr 45161

This theorem is referenced by:  isisomgr  45164  isomgreqve  45165  isomushgr  45166  isomgrsym  45176  isomgrtr  45179  ushrisomgr  45181