Theorem isomgrtr 42557

Description: The isomorphy relation is transitive for hypergraphs. (Contributed by AV, 5-Dec-2022.)

Assertion

Ref	Expression
isomgrtr	⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → ((𝐴 IsomGr 𝐵 ∧ 𝐵 IsomGr 𝐶) → 𝐴 IsomGr 𝐶))

Proof of Theorem isomgrtr

Dummy variables 𝑖 𝑗 𝑘 𝑓 𝑔 𝑣 𝑤 𝑒 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2825	. . . . 5 ⊢ (Vtx‘𝐴) = (Vtx‘𝐴)
2		eqid 2825	. . . . 5 ⊢ (Vtx‘𝐵) = (Vtx‘𝐵)
3		eqid 2825	. . . . 5 ⊢ (iEdg‘𝐴) = (iEdg‘𝐴)
4		eqid 2825	. . . . 5 ⊢ (iEdg‘𝐵) = (iEdg‘𝐵)
5	1, 2, 3, 4	isomgr 42541	. . . 4 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))))
6	5	3adant3 1168	. . 3 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))))
7		eqid 2825	. . . . 5 ⊢ (Vtx‘𝐶) = (Vtx‘𝐶)
8		eqid 2825	. . . . 5 ⊢ (iEdg‘𝐶) = (iEdg‘𝐶)
9	2, 7, 4, 8	isomgr 42541	. . . 4 ⊢ ((𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → (𝐵 IsomGr 𝐶 ↔ ∃𝑣(𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))))))
10	9	3adant1 1166	. . 3 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → (𝐵 IsomGr 𝐶 ↔ ∃𝑣(𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))))))
11	6, 10	anbi12d 626	. 2 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → ((𝐴 IsomGr 𝐵 ∧ 𝐵 IsomGr 𝐶) ↔ (∃𝑓(𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ ∃𝑣(𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))))))
12		vex 3417	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
13		vex 3417	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
14	12, 13	coex 7380	. . . . . . . . . 10 ⊢ (𝑣 ∘ 𝑓) ∈ V
15	14	a1i 11	. . . . . . . . 9 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) ∧ (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))))) → (𝑣 ∘ 𝑓) ∈ V)
16		simpl 476	. . . . . . . . . . 11 ⊢ ((𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶))
17		simprl 789	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) → 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵))
18		f1oco 6400	. . . . . . . . . . 11 ⊢ ((𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) → (𝑣 ∘ 𝑓):(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶))
19	16, 17, 18	syl2anr 592	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) ∧ (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))))) → (𝑣 ∘ 𝑓):(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶))
20		vex 3417	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑤 ∈ V
21		vex 3417	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑔 ∈ V
22	20, 21	coex 7380	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∘ 𝑔) ∈ V
23	22	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → (𝑤 ∘ 𝑔) ∈ V)
24		simpl 476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))) → 𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶))
25		simprl 789	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) → 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵))
26		f1oco 6400	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵)) → (𝑤 ∘ 𝑔):dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶))
27	24, 25, 26	syl2anr 592	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → (𝑤 ∘ 𝑔):dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶))
28		isomgrtrlem 42556	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘((𝑤 ∘ 𝑔)‘𝑗)))
29	27, 28	jca 509	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → ((𝑤 ∘ 𝑔):dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘((𝑤 ∘ 𝑔)‘𝑗))))
30		f1oeq1 6367	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = (𝑤 ∘ 𝑔) → (ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ↔ (𝑤 ∘ 𝑔):dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶)))
31		fveq1 6432	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = (𝑤 ∘ 𝑔) → (ℎ‘𝑗) = ((𝑤 ∘ 𝑔)‘𝑗))
32	31	fveq2d 6437	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ = (𝑤 ∘ 𝑔) → ((iEdg‘𝐶)‘(ℎ‘𝑗)) = ((iEdg‘𝐶)‘((𝑤 ∘ 𝑔)‘𝑗)))
33	32	eqeq2d 2835	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ = (𝑤 ∘ 𝑔) → (((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)) ↔ ((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘((𝑤 ∘ 𝑔)‘𝑗))))
34	33	ralbidv 3195	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = (𝑤 ∘ 𝑔) → (∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)) ↔ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘((𝑤 ∘ 𝑔)‘𝑗))))
35	30, 34	anbi12d 626	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = (𝑤 ∘ 𝑔) → ((ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))) ↔ ((𝑤 ∘ 𝑔):dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘((𝑤 ∘ 𝑔)‘𝑗)))))
36	23, 29, 35	elabd 3573	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))))
37	36	ex 403	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) → ((𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))) → ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))))
38	37	exlimdv 2034	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) → (∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))) → ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))))
39	38	ex 403	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) → ((𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) → (∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))) → ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))))))
40	39	exlimdv 2034	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) → (∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) → (∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))) → ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))))))
41	40	3exp 1154	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) → (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) → (∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) → (∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))) → ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))))))))
42	41	com34 91	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) → (∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) → (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) → (∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))) → ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))))))))
43	42	imp32 411	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) → (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) → (∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))) → ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))))))
44	43	imp32 411	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) ∧ (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))))) → ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))))
45	19, 44	jca 509	. . . . . . . . 9 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) ∧ (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))))) → ((𝑣 ∘ 𝑓):(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))))
46		f1oeq1 6367	. . . . . . . . . 10 ⊢ (𝑒 = (𝑣 ∘ 𝑓) → (𝑒:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶) ↔ (𝑣 ∘ 𝑓):(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶)))
47		imaeq1 5702	. . . . . . . . . . . . . 14 ⊢ (𝑒 = (𝑣 ∘ 𝑓) → (𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)))
48	47	eqeq1d 2827	. . . . . . . . . . . . 13 ⊢ (𝑒 = (𝑣 ∘ 𝑓) → ((𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)) ↔ ((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))))
49	48	ralbidv 3195	. . . . . . . . . . . 12 ⊢ (𝑒 = (𝑣 ∘ 𝑓) → (∀𝑗 ∈ dom (iEdg‘𝐴)(𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)) ↔ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))))
50	49	anbi2d 624	. . . . . . . . . . 11 ⊢ (𝑒 = (𝑣 ∘ 𝑓) → ((ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)(𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))) ↔ (ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))))
51	50	exbidv 2022	. . . . . . . . . 10 ⊢ (𝑒 = (𝑣 ∘ 𝑓) → (∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)(𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))) ↔ ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))))
52	46, 51	anbi12d 626	. . . . . . . . 9 ⊢ (𝑒 = (𝑣 ∘ 𝑓) → ((𝑒:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)(𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))) ↔ ((𝑣 ∘ 𝑓):(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))))))
53	15, 45, 52	elabd 3573	. . . . . . . 8 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) ∧ (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))))) → ∃𝑒(𝑒:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)(𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗)))))
54	1, 7, 3, 8	isomgr 42541	. . . . . . . . . 10 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → (𝐴 IsomGr 𝐶 ↔ ∃𝑒(𝑒:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)(𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))))))
55	54	3adant2 1167	. . . . . . . . 9 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → (𝐴 IsomGr 𝐶 ↔ ∃𝑒(𝑒:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)(𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))))))
56	55	ad2antrr 719	. . . . . . . 8 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) ∧ (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))))) → (𝐴 IsomGr 𝐶 ↔ ∃𝑒(𝑒:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐶) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑗 ∈ dom (iEdg‘𝐴)(𝑒 “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘(ℎ‘𝑗))))))
57	53, 56	mpbird 249	. . . . . . 7 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) ∧ (𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))))) → 𝐴 IsomGr 𝐶)
58	57	ex 403	. . . . . 6 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) → ((𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → 𝐴 IsomGr 𝐶))
59	58	exlimdv 2034	. . . . 5 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))) → (∃𝑣(𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → 𝐴 IsomGr 𝐶))
60	59	ex 403	. . . 4 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → ((𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) → (∃𝑣(𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → 𝐴 IsomGr 𝐶)))
61	60	exlimdv 2034	. . 3 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → (∃𝑓(𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) → (∃𝑣(𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → 𝐴 IsomGr 𝐶)))
62	61	impd 400	. 2 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → ((∃𝑓(𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ ∃𝑣(𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶) ∧ ∃𝑤(𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘))))) → 𝐴 IsomGr 𝐶))
63	11, 62	sylbid 232	1 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → ((𝐴 IsomGr 𝐵 ∧ 𝐵 IsomGr 𝐶) → 𝐴 IsomGr 𝐶))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1113   = wceq 1658  ∃wex 1880   ∈ wcel 2166  ∀wral 3117  Vcvv 3414   class class class wbr 4873  dom cdm 5342   “ cima 5345   ∘ ccom 5346  –1-1-onto→wf1o 6122  ‘cfv 6123  Vtxcvtx 26294  iEdgciedg 26295  UHGraphcuhgr 26354   IsomGr cisomgr 42537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isomgr 42539

This theorem is referenced by: (None)