grimco - Mathbox for Alexander van der Vekens

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Theorem grimco 48471

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 2761	. . . 4 ⊢ (Vtx‘𝑇) = (Vtx‘𝑇)
2		eqid 2761	. . . 4 ⊢ (Vtx‘𝑈) = (Vtx‘𝑈)
3		eqid 2761	. . . 4 ⊢ (iEdg‘𝑇) = (iEdg‘𝑇)
4		eqid 2761	. . . 4 ⊢ (iEdg‘𝑈) = (iEdg‘𝑈)
5	1, 2, 3, 4	grimprop 48465	. . 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 2761	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
7		eqid 2761	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
8	6, 1, 7, 3	grimprop 48465	. . 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 6824	. . . . 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 757	. . . 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 3457	. . . . . . . . . . . . 13 ⊢ 𝑓 ∈ V
12		vex 3457	. . . . . . . . . . . . 13 ⊢ 𝑔 ∈ V
13	11, 12	coex 7905	. . . . . . . . . . . 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 6824	. . . . . . . . . . . . . . . . . 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 25	. . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 410	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 6866	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑖 → ((iEdg‘𝑇)‘(𝑔‘𝑦)) = ((iEdg‘𝑇)‘(𝑔‘𝑖)))
23		fveq2 6861	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑖 → ((iEdg‘𝑆)‘𝑦) = ((iEdg‘𝑆)‘𝑖))
24	23	imaeq2d 6044	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑖 → (𝐺 “ ((iEdg‘𝑆)‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑖)))
25	22, 24	eqeq12d 2777	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑖 → (((iEdg‘𝑇)‘(𝑔‘𝑦)) = (𝐺 “ ((iEdg‘𝑆)‘𝑦)) ↔ ((iEdg‘𝑇)‘(𝑔‘𝑖)) = (𝐺 “ ((iEdg‘𝑆)‘𝑖))))
26	25	rspcv 3576	. . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 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 6800	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑔:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) → 𝑔:dom (iEdg‘𝑆)⟶dom (iEdg‘𝑇))
30	29	adantl 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)) → 𝑔:dom (iEdg‘𝑆)⟶dom (iEdg‘𝑇))
31	30	ffvelcdmda 7059	. . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . 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 6866	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = (𝑔‘𝑖) → ((iEdg‘𝑈)‘(𝑓‘𝑥)) = ((iEdg‘𝑈)‘(𝑓‘(𝑔‘𝑖))))
34		fveq2 6861	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = (𝑔‘𝑖) → ((iEdg‘𝑇)‘𝑥) = ((iEdg‘𝑇)‘(𝑔‘𝑖)))
35	34	imaeq2d 6044	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = (𝑔‘𝑖) → (𝐹 “ ((iEdg‘𝑇)‘𝑥)) = (𝐹 “ ((iEdg‘𝑇)‘(𝑔‘𝑖))))
36	33, 35	eqeq12d 2777	. . . . . . . . . . . . . . . . . . . . . . . 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 3576	. . . . . . . . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6962	. . . . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . 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 6865	. . . . . . . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . . . . . . . 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 47	. . . . . . . . . . . . . . . . . . . . 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 458	. . . . . . . . . . . . . . . . . . . 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 6040	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((iEdg‘𝑇)‘(𝑔‘𝑖)) = (𝐺 “ ((iEdg‘𝑆)‘𝑖)) → (𝐹 “ ((iEdg‘𝑇)‘(𝑔‘𝑖))) = (𝐹 “ (𝐺 “ ((iEdg‘𝑆)‘𝑖))))
52		imaco 6232	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖)) = (𝐹 “ (𝐺 “ ((iEdg‘𝑆)‘𝑖)))
53	51, 52	eqtr4di 2814	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((iEdg‘𝑇)‘(𝑔‘𝑖)) = (𝐺 “ ((iEdg‘𝑆)‘𝑖)) → (𝐹 “ ((iEdg‘𝑇)‘(𝑔‘𝑖))) = ((𝐹 ∘ 𝐺) “ ((iEdg‘𝑆)‘𝑖)))
54	50, 53	sylan9eq 2816	. . . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . . . 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 47	. . . . . . . . . . . . . . . . 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 423	. . . . . . . . . . . . . . . 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 95	. . . . . . . . . . . . . . 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 457	. . . . . . . . . . . . . 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 422	. . . . . . . . . . . . 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 3152	. . . . . . . . . . . 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 519	. . . . . . . . . . 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 6788	. . . . . . . . . . . 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 6860	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝑓 ∘ 𝑔) → (𝑗‘𝑖) = ((𝑓 ∘ 𝑔)‘𝑖))
65	64	fveqeq2d 6869	. . . . . . . . . . . . 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 3184	. . . . . . . . . . . 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 641	. . . . . . . . . . 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 3556	. . . . . . . . . 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 424	. . . . . . . . 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 1952	. . . . . . . 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 457	. . . . . . 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 86	. . . . . 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 1952	. . . . 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 421	. . . 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 519	. . 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 605	. 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 48462	. . . . . 6 ⊢ Rel dom GraphIso
78	77	ovrcl 7431	. . . . 5 ⊢ (𝐺 ∈ (𝑆 GraphIso 𝑇) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
79	78	simpld 498	. . . 4 ⊢ (𝐺 ∈ (𝑆 GraphIso 𝑇) → 𝑆 ∈ V)
80	79	adantl 485	. . 3 ⊢ ((𝐹 ∈ (𝑇 GraphIso 𝑈) ∧ 𝐺 ∈ (𝑆 GraphIso 𝑇)) → 𝑆 ∈ V)
81	77	ovrcl 7431	. . . . 5 ⊢ (𝐹 ∈ (𝑇 GraphIso 𝑈) → (𝑇 ∈ V ∧ 𝑈 ∈ V))
82	81	simprd 499	. . . 4 ⊢ (𝐹 ∈ (𝑇 GraphIso 𝑈) → 𝑈 ∈ V)
83	82	adantr 484	. . 3 ⊢ ((𝐹 ∈ (𝑇 GraphIso 𝑈) ∧ 𝐺 ∈ (𝑆 GraphIso 𝑇)) → 𝑈 ∈ V)
84		coexg 7904	. . 3 ⊢ ((𝐹 ∈ (𝑇 GraphIso 𝑈) ∧ 𝐺 ∈ (𝑆 GraphIso 𝑇)) → (𝐹 ∘ 𝐺) ∈ V)
85	6, 2, 7, 4	isgrim 48464	. . 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 1389	. 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 259	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 208 ∧ wa 399 = wceq 1559 ∃wex 1798 ∈ wcel 2141 ∀wral 3075 Vcvv 3453 dom cdm 5643 “ cima 5646 ∘ ccom 5647 ⟶wf 6511 –1-1-onto→wf1o 6514 ‘cfv 6515 (class class class)co 7390 Vtxcvtx 29153 iEdgciedg 29154 GraphIso cgrim 48457

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-sbc 3743 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-br 5098 df-opab 5160 df-id 5538 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-ov 7393 df-oprab 7394 df-mpo 7395 df-map 8803 df-grim 48460

This theorem is referenced by: grictr 48505

W3C validator