grimco - Mathbox for Alexander van der Vekens

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Theorem grimco 48637

Description: The composition of graph isomorphisms is a graph isomorphism. (Contributed by AV, 3-May-2025.)

**Assertion**
Ref	Expression
grimco	⊢ ((𝐹 ∈ (𝑇 GraphIso 𝑈) ∧ 𝐺 ∈ (𝑆 GraphIso 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GraphIso 𝑈))

Proof of Theorem grimco Dummy variables 𝑓 𝑔 𝑖 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . 4 ⊢ (Vtx‘𝑇) = (Vtx‘𝑇)
2		eqid 2763	. . . 4 ⊢ (Vtx‘𝑈) = (Vtx‘𝑈)
3		eqid 2763	. . . 4 ⊢ (iEdg‘𝑇) = (iEdg‘𝑇)
4		eqid 2763	. . . 4 ⊢ (iEdg‘𝑈) = (iEdg‘𝑈)
5	1, 2, 3, 4	grimprop 48631	. . 3 ⊢ (𝐹 ∈ (𝑇 GraphIso 𝑈) → (𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ ∃𝑓(𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥)))))
6		eqid 2763	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
7		eqid 2763	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
8	6, 1, 7, 3	grimprop 48631	. . 3 ⊢ (𝐺 ∈ (𝑆 GraphIso 𝑇) → (𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ ∃𝑔(𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦)))))
9		f1oco 6846	. . . . 5 ⊢ ((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) → (𝐹 ∘ 𝐺):(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑈))
10	9	ad2ant2r 759	. . . 4 ⊢ (((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ ∃𝑓(𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥)))) ∧ (𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ ∃𝑔(𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦))))) → (𝐹 ∘ 𝐺):(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑈))
11		vex 3459	. . . . . . . . . . . . 13 ⊢ 𝑓 ∈ V
12		vex 3459	. . . . . . . . . . . . 13 ⊢ 𝑔 ∈ V
13	11, 12	coex 7928	. . . . . . . . . . . 12 ⊢ (𝑓 ∘ 𝑔) ∈ V
14	13	a1i 11	. . . . . . . . . . 11 ⊢ (((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ ((𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦))) ∧ (𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥))))) → (𝑓 ∘ 𝑔) ∈ V)
15		f1oco 6846	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) → (𝑓 ∘ 𝑔):dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈))
16	15	a1d 26	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) → (∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥)) → (𝑓 ∘ 𝑔):dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈)))
17	16	expcom 418	. . . . . . . . . . . . . . . 16 ⊢ (𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) → (𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) → (∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥)) → (𝑓 ∘ 𝑔):dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈))))
18	17	impd 415	. . . . . . . . . . . . . . 15 ⊢ (𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) → ((𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥))) → (𝑓 ∘ 𝑔):dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈)))
19	18	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦))) → ((𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥))) → (𝑓 ∘ 𝑔):dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈)))
20	19	imp 411	. . . . . . . . . . . . 13 ⊢ (((𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦))) ∧ (𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥)))) → (𝑓 ∘ 𝑔):dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈))
21	20	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ ((𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦))) ∧ (𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥))))) → (𝑓 ∘ 𝑔):dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈))
22		2fveq3 6888	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑖 → ((iEdg‘𝑇)‘(𝑔‘𝑦)) = ((iEdg‘𝑇)‘(𝑔‘𝑖)))
23		fveq2 6883	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑖 → ((iEdg‘𝑆)‘𝑦) = ((iEdg‘𝑆)‘𝑖))
24	23	imaeq2d 6064	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑖 → (𝐺 “ ((iEdg‘𝑆)‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑖)))
25	22, 24	eqeq12d 2779	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑖 → (((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦)) ↔ ((iEdg‘𝑇)‘(𝑔‘𝑖)) = (𝐺 “ ((iEdg‘𝑆)‘𝑖))))
26	25	rspcv 3578	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ dom (iEdg‘𝑆) → (∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦)) → ((iEdg‘𝑇)‘(𝑔‘𝑖)) = (𝐺 “ ((iEdg‘𝑆)‘𝑖))))
27	26	adantl 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → (∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦)) → ((iEdg‘𝑇)‘(𝑔‘𝑖)) = (𝐺 “ ((iEdg‘𝑆)‘𝑖))))
28	27	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) ∧ (𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥)))) → (∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦)) → ((iEdg‘𝑇)‘(𝑔‘𝑖)) = (𝐺 “ ((iEdg‘𝑆)‘𝑖))))
29		f1of 6822	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) → 𝑔:dom (iEdg‘𝑆)⟶dom (iEdg‘𝑇))
30	29	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) → 𝑔:dom (iEdg‘𝑆)⟶dom (iEdg‘𝑇))
31	30	ffvelcdmda 7081	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → (𝑔‘𝑖) ∈ dom (iEdg‘𝑇))
32	31	adantr 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) ∧ 𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈)) → (𝑔‘𝑖) ∈ dom (iEdg‘𝑇))
33		2fveq3 6888	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = (𝑔‘𝑖) → ((iEdg‘𝑈)‘(𝑓‘𝑥)) = ((iEdg‘𝑈)‘(𝑓‘(𝑔‘𝑖))))
34		fveq2 6883	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = (𝑔‘𝑖) → ((iEdg‘𝑇)‘𝑥) = ((iEdg‘𝑇)‘(𝑔‘𝑖)))
35	34	imaeq2d 6064	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = (𝑔‘𝑖) → (𝐹 “ ((iEdg‘𝑇)‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘(𝑔‘𝑖))))
36	33, 35	eqeq12d 2779	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = (𝑔‘𝑖) → (((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥)) ↔ ((iEdg‘𝑈)‘(𝑓‘(𝑔‘𝑖))) = (𝐹 “ ((iEdg‘𝑇)‘(𝑔‘𝑖)))))
37	36	rspcv 3578	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑔‘𝑖) ∈ dom (iEdg‘𝑇) → (∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥)) → ((iEdg‘𝑈)‘(𝑓‘(𝑔‘𝑖))) = (𝐹 “ ((iEdg‘𝑇)‘(𝑔‘𝑖)))))
38	32, 37	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) ∧ 𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈)) → (∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥)) → ((iEdg‘𝑈)‘(𝑓‘(𝑔‘𝑖))) = (𝐹 “ ((iEdg‘𝑇)‘(𝑔‘𝑖)))))
39	30	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → 𝑔:dom (iEdg‘𝑆)⟶dom (iEdg‘𝑇))
40	39	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) ∧ 𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈)) → 𝑔:dom (iEdg‘𝑆)⟶dom (iEdg‘𝑇))
41		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) → 𝑖 ∈ dom (iEdg‘𝑆))
42	41	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) ∧ 𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈)) → 𝑖 ∈ dom (iEdg‘𝑆))
43	40, 42	fvco3d 6984	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) ∧ 𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈)) → ((𝑓 ∘ 𝑔)‘𝑖) = (𝑓‘(𝑔‘𝑖)))
44	43	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) ∧ 𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈)) ∧ ((iEdg‘𝑈)‘(𝑓‘(𝑔‘𝑖))) = (𝐹 “ ((iEdg‘𝑇)‘(𝑔‘𝑖)))) → ((𝑓 ∘ 𝑔)‘𝑖) = (𝑓‘(𝑔‘𝑖)))
45	44	fveq2d 6887	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) ∧ 𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈)) ∧ ((iEdg‘𝑈)‘(𝑓‘(𝑔‘𝑖))) = (𝐹 “ ((iEdg‘𝑇)‘(𝑔‘𝑖)))) → ((iEdg‘𝑈)‘((𝑓 ∘ 𝑔)‘𝑖)) = ((iEdg‘𝑈)‘(𝑓‘(𝑔‘𝑖))))
46		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) ∧ 𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈)) ∧ ((iEdg‘𝑈)‘(𝑓‘(𝑔‘𝑖))) = (𝐹 “ ((iEdg‘𝑇)‘(𝑔‘𝑖)))) → ((iEdg‘𝑈)‘(𝑓‘(𝑔‘𝑖))) = (𝐹 “ ((iEdg‘𝑇)‘(𝑔‘𝑖))))
47	45, 46	eqtrd 2798	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) ∧ 𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈)) ∧ ((iEdg‘𝑈)‘(𝑓‘(𝑔‘𝑖))) = (𝐹 “ ((iEdg‘𝑇)‘(𝑔‘𝑖)))) → ((iEdg‘𝑈)‘((𝑓 ∘ 𝑔)‘𝑖)) = (𝐹 “ ((iEdg‘𝑇)‘(𝑔‘𝑖))))
48	47	ex 417	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) ∧ 𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈)) → (((iEdg‘𝑈)‘(𝑓‘(𝑔‘𝑖))) = (𝐹 “ ((iEdg‘𝑇)‘(𝑔‘𝑖))) → ((iEdg‘𝑈)‘((𝑓 ∘ 𝑔)‘𝑖)) = (𝐹 “ ((iEdg‘𝑇)‘(𝑔‘𝑖)))))
49	38, 48	syld 48	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) ∧ 𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈)) → (∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥)) → ((iEdg‘𝑈)‘((𝑓 ∘ 𝑔)‘𝑖)) = (𝐹 “ ((iEdg‘𝑇)‘(𝑔‘𝑖)))))
50	49	impr 459	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) ∧ (𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥)))) → ((iEdg‘𝑈)‘((𝑓 ∘ 𝑔)‘𝑖)) = (𝐹 “ ((iEdg‘𝑇)‘(𝑔‘𝑖))))
51		imaeq2 6060	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((iEdg‘𝑇)‘(𝑔‘𝑖)) = (𝐺 “ ((iEdg‘𝑆)‘𝑖)) → (𝐹 “ ((iEdg‘𝑇)‘(𝑔‘𝑖))) = (𝐹 “ (𝐺 “ ((iEdg‘𝑆)‘𝑖))))
52		imaco 6254	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖)) = (𝐹 “ (𝐺 “ ((iEdg‘𝑆)‘𝑖)))
53	51, 52	eqtr4di 2816	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((iEdg‘𝑇)‘(𝑔‘𝑖)) = (𝐺 “ ((iEdg‘𝑆)‘𝑖)) → (𝐹 “ ((iEdg‘𝑇)‘(𝑔‘𝑖))) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖)))
54	50, 53	sylan9eq 2818	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) ∧ (𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥)))) ∧ ((iEdg‘𝑇)‘(𝑔‘𝑖)) = (𝐺 “ ((iEdg‘𝑆)‘𝑖))) → ((iEdg‘𝑈)‘((𝑓 ∘ 𝑔)‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖)))
55	54	ex 417	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) ∧ (𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥)))) → (((iEdg‘𝑇)‘(𝑔‘𝑖)) = (𝐺 “ ((iEdg‘𝑆)‘𝑖)) → ((iEdg‘𝑈)‘((𝑓 ∘ 𝑔)‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖))))
56	28, 55	syld 48	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑖 ∈ dom (iEdg‘𝑆)) ∧ (𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥)))) → (∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦)) → ((iEdg‘𝑈)‘((𝑓 ∘ 𝑔)‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖))))
57	56	exp31 424	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) → (𝑖 ∈ dom (iEdg‘𝑆) → ((𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥))) → (∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦)) → ((iEdg‘𝑈)‘((𝑓 ∘ 𝑔)‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖))))))
58	57	com24 96	. . . . . . . . . . . . . . 15 ⊢ (((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) → (∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦)) → ((𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥))) → (𝑖 ∈ dom (iEdg‘𝑆) → ((iEdg‘𝑈)‘((𝑓 ∘ 𝑔)‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖))))))
59	58	expimpd 458	. . . . . . . . . . . . . 14 ⊢ ((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) → ((𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦))) → ((𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥))) → (𝑖 ∈ dom (iEdg‘𝑆) → ((iEdg‘𝑈)‘((𝑓 ∘ 𝑔)‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖))))))
60	59	imp32 423	. . . . . . . . . . . . 13 ⊢ (((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ ((𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦))) ∧ (𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥))))) → (𝑖 ∈ dom (iEdg‘𝑆) → ((iEdg‘𝑈)‘((𝑓 ∘ 𝑔)‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖))))
61	60	ralrimiv 3156	. . . . . . . . . . . 12 ⊢ (((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ ((𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦))) ∧ (𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥))))) → ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑈)‘((𝑓 ∘ 𝑔)‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖)))
62	21, 61	jca 520	. . . . . . . . . . 11 ⊢ (((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ ((𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦))) ∧ (𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥))))) → ((𝑓 ∘ 𝑔):dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑈)‘((𝑓 ∘ 𝑔)‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖))))
63		f1oeq1 6810	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝑓 ∘ 𝑔) → (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈) ↔ (𝑓 ∘ 𝑔):dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈)))
64		fveq1 6882	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝑓 ∘ 𝑔) → (𝑗‘𝑖) = ((𝑓 ∘ 𝑔)‘𝑖))
65	64	fveqeq2d 6891	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑓 ∘ 𝑔) → (((iEdg‘𝑈)‘(𝑗‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖)) ↔ ((iEdg‘𝑈)‘((𝑓 ∘ 𝑔)‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖))))
66	65	ralbidv 3188	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝑓 ∘ 𝑔) → (∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑈)‘(𝑗‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖)) ↔ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑈)‘((𝑓 ∘ 𝑔)‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖))))
67	63, 66	anbi12d 643	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑓 ∘ 𝑔) → ((𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑈)‘(𝑗‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖))) ↔ ((𝑓 ∘ 𝑔):dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑈)‘((𝑓 ∘ 𝑔)‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖)))))
68	14, 62, 67	spcedv 3558	. . . . . . . . . 10 ⊢ (((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ ((𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦))) ∧ (𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥))))) → ∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑈)‘(𝑗‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖))))
69	68	exp32 425	. . . . . . . . 9 ⊢ ((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) → ((𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦))) → ((𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥))) → ∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑈)‘(𝑗‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖))))))
70	69	exlimdv 1963	. . . . . . . 8 ⊢ ((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ 𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) → (∃𝑔(𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦))) → ((𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥))) → ∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑈)‘(𝑗‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖))))))
71	70	expimpd 458	. . . . . . 7 ⊢ (𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) → ((𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ ∃𝑔(𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦)))) → ((𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥))) → ∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑈)‘(𝑗‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖))))))
72	71	com23 87	. . . . . 6 ⊢ (𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) → ((𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥))) → ((𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ ∃𝑔(𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦)))) → ∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑈)‘(𝑗‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖))))))
73	72	exlimdv 1963	. . . . 5 ⊢ (𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) → (∃𝑓(𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥))) → ((𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ ∃𝑔(𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦)))) → ∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑈)‘(𝑗‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖))))))
74	73	imp31 422	. . . 4 ⊢ (((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ ∃𝑓(𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥)))) ∧ (𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ ∃𝑔(𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦))))) → ∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑈)‘(𝑗‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖))))
75	10, 74	jca 520	. . 3 ⊢ (((𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑈) ∧ ∃𝑓(𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑈)‘(𝑓‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘𝑥)))) ∧ (𝐺:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ ∃𝑔(𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑦 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦))))) → ((𝐹 ∘ 𝐺):(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑈) ∧ ∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑈)‘(𝑗‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖)))))
76	5, 8, 75	syl2an 607	. 2 ⊢ ((𝐹 ∈ (𝑇 GraphIso 𝑈) ∧ 𝐺 ∈ (𝑆 GraphIso 𝑇)) → ((𝐹 ∘ 𝐺):(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑈) ∧ ∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑈)‘(𝑗‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖)))))
77		grimdmrel 48628	. . . . . 6 ⊢ Rel dom GraphIso
78	77	ovrcl 7453	. . . . 5 ⊢ (𝐺 ∈ (𝑆 GraphIso 𝑇) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
79	78	simpld 499	. . . 4 ⊢ (𝐺 ∈ (𝑆 GraphIso 𝑇) → 𝑆 ∈ V)
80	79	adantl 486	. . 3 ⊢ ((𝐹 ∈ (𝑇 GraphIso 𝑈) ∧ 𝐺 ∈ (𝑆 GraphIso 𝑇)) → 𝑆 ∈ V)
81	77	ovrcl 7453	. . . . 5 ⊢ (𝐹 ∈ (𝑇 GraphIso 𝑈) → (𝑇 ∈ V ∧ 𝑈 ∈ V))
82	81	simprd 500	. . . 4 ⊢ (𝐹 ∈ (𝑇 GraphIso 𝑈) → 𝑈 ∈ V)
83	82	adantr 485	. . 3 ⊢ ((𝐹 ∈ (𝑇 GraphIso 𝑈) ∧ 𝐺 ∈ (𝑆 GraphIso 𝑇)) → 𝑈 ∈ V)
84		coexg 7927	. . 3 ⊢ ((𝐹 ∈ (𝑇 GraphIso 𝑈) ∧ 𝐺 ∈ (𝑆 GraphIso 𝑇)) → (𝐹 ∘ 𝐺) ∈ V)
85	6, 2, 7, 4	isgrim 48630	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝑈 ∈ V ∧ (𝐹 ∘ 𝐺) ∈ V) → ((𝐹 ∘ 𝐺) ∈ (𝑆 GraphIso 𝑈) ↔ ((𝐹 ∘ 𝐺):(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑈) ∧ ∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑈)‘(𝑗‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖))))))
86	80, 83, 84, 85	syl3anc 1398	. 2 ⊢ ((𝐹 ∈ (𝑇 GraphIso 𝑈) ∧ 𝐺 ∈ (𝑆 GraphIso 𝑇)) → ((𝐹 ∘ 𝐺) ∈ (𝑆 GraphIso 𝑈) ↔ ((𝐹 ∘ 𝐺):(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑈) ∧ ∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑈) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑈)‘(𝑗‘𝑖)) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖))))))
87	76, 86	mpbird 260	1 ⊢ ((𝐹 ∈ (𝑇 GraphIso 𝑈) ∧ 𝐺 ∈ (𝑆 GraphIso 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GraphIso 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∃wex 1809 ∈ wcel 2143 ∀wral 3079 Vcvv 3455 dom cdm 5663 “ cima 5666 ∘ ccom 5667 ⟶wf 6534 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7412 Vtxcvtx 29327 iEdgciedg 29328 GraphIso cgrim 48623

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3746 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-br 5111 df-opab 5175 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7415 df-oprab 7416 df-mpo 7417 df-map 8827 df-grim 48626

This theorem is referenced by: grictr 48671

W3C validator