Theorem isomgr 42746

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 2779	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑓 = 𝑓)
2		fveq2 6448	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (Vtx‘𝑥) = (Vtx‘𝐴))
3	2	adantr 474	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (Vtx‘𝑥) = (Vtx‘𝐴))
4		isomgr.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐴)
5	3, 4	syl6eqr 2832	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (Vtx‘𝑥) = 𝑉)
6		fveq2 6448	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (Vtx‘𝑦) = (Vtx‘𝐵))
7	6	adantl 475	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (Vtx‘𝑦) = (Vtx‘𝐵))
8		isomgr.w	. . . . . 6 ⊢ 𝑊 = (Vtx‘𝐵)
9	7, 8	syl6eqr 2832	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (Vtx‘𝑦) = 𝑊)
10	1, 5, 9	f1oeq123d 6388	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑓:(Vtx‘𝑥)–1-1-onto→(Vtx‘𝑦) ↔ 𝑓:𝑉–1-1-onto→𝑊))
11		eqidd 2779	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑔 = 𝑔)
12		fveq2 6448	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (iEdg‘𝑥) = (iEdg‘𝐴))
13	12	adantr 474	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (iEdg‘𝑥) = (iEdg‘𝐴))
14		isomgr.i	. . . . . . . . 9 ⊢ 𝐼 = (iEdg‘𝐴)
15	13, 14	syl6eqr 2832	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (iEdg‘𝑥) = 𝐼)
16	15	dmeqd 5573	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → dom (iEdg‘𝑥) = dom 𝐼)
17		fveq2 6448	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (iEdg‘𝑦) = (iEdg‘𝐵))
18	17	adantl 475	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (iEdg‘𝑦) = (iEdg‘𝐵))
19		isomgr.j	. . . . . . . . 9 ⊢ 𝐽 = (iEdg‘𝐵)
20	18, 19	syl6eqr 2832	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (iEdg‘𝑦) = 𝐽)
21	20	dmeqd 5573	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → dom (iEdg‘𝑦) = dom 𝐽)
22	11, 16, 21	f1oeq123d 6388	. . . . . 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 6450	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((iEdg‘𝑥)‘𝑖) = (𝐼‘𝑖))
24	23	imaeq2d 5722	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑓 “ ((iEdg‘𝑥)‘𝑖)) = (𝑓 “ (𝐼‘𝑖)))
25	20	fveq1d 6450	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((iEdg‘𝑦)‘(𝑔‘𝑖)) = (𝐽‘(𝑔‘𝑖)))
26	24, 25	eqeq12d 2793	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔‘𝑖)) ↔ (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))
27	16, 26	raleqbidv 3326	. . . . . 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 624	. . . . 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 1964	. . . 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 624	. . 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 1964	. 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 42744	. 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 5228	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 198   ∧ wa 386   = wceq 1601  ∃wex 1823   ∈ wcel 2107  ∀wral 3090   class class class wbr 4888  dom cdm 5357   “ cima 5360  –1-1-onto→wf1o 6136  ‘cfv 6137  Vtxcvtx 26348  iEdgciedg 26349   IsomGr cisomgr 42742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-isomgr 42744

This theorem is referenced by:  isisomgr  42747  isomgreqve  42748  isomushgr  42749  isomgrsym  42759  isomgrtr  42762  ushrisomgr  42764