Theorem ismtyres 32777

Description: A restriction of an isometry is an isometry. The condition 𝐴 ⊆ 𝑋 is not necessary but makes the proof easier. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)

Hypotheses

Ref	Expression
ismtyres.2	⊢ 𝐵 = (𝐹 “ 𝐴)
ismtyres.3	⊢ 𝑆 = (𝑀 ↾ (𝐴 × 𝐴))
ismtyres.4	⊢ 𝑇 = (𝑁 ↾ (𝐵 × 𝐵))

Assertion

Ref	Expression
ismtyres	⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → (𝐹 ↾ 𝐴) ∈ (𝑆 Ismty 𝑇))

Proof of Theorem ismtyres

Dummy variables 𝑣 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isismty 32770	. . . . . 6 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)))))
2	1	simprbda 651	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → 𝐹:𝑋–1-1-onto→𝑌)
3	2	adantrr 749	. . . 4 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝐹:𝑋–1-1-onto→𝑌)
4		f1of1 6049	. . . 4 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–1-1→𝑌)
5	3, 4	syl 17	. . 3 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝐹:𝑋–1-1→𝑌)
6		simprr 792	. . 3 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝐴 ⊆ 𝑋)
7		f1ores 6064	. . 3 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴))
8	5, 6, 7	syl2anc 691	. 2 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴))
9	1	biimpa 500	. . . 4 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))))
10	9	adantrr 749	. . 3 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))))
11		ssel 3562	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑋 → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑋))
12		ssel 3562	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑋 → (𝑣 ∈ 𝐴 → 𝑣 ∈ 𝑋))
13	11, 12	anim12d 584	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝑋 → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)))
14	13	imp 444	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝑋 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋))
15		oveq1 6556	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → (𝑥𝑀𝑦) = (𝑢𝑀𝑦))
16		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (𝐹‘𝑥) = (𝐹‘𝑢))
17	16	oveq1d 6564	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) = ((𝐹‘𝑢)𝑁(𝐹‘𝑦)))
18	15, 17	eqeq12d 2625	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → ((𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) ↔ (𝑢𝑀𝑦) = ((𝐹‘𝑢)𝑁(𝐹‘𝑦))))
19		oveq2 6557	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑣 → (𝑢𝑀𝑦) = (𝑢𝑀𝑣))
20		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑣 → (𝐹‘𝑦) = (𝐹‘𝑣))
21	20	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑣 → ((𝐹‘𝑢)𝑁(𝐹‘𝑦)) = ((𝐹‘𝑢)𝑁(𝐹‘𝑣)))
22	19, 21	eqeq12d 2625	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑣 → ((𝑢𝑀𝑦) = ((𝐹‘𝑢)𝑁(𝐹‘𝑦)) ↔ (𝑢𝑀𝑣) = ((𝐹‘𝑢)𝑁(𝐹‘𝑣))))
23	18, 22	rspc2v 3293	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) → (𝑢𝑀𝑣) = ((𝐹‘𝑢)𝑁(𝐹‘𝑣))))
24	14, 23	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ 𝑋 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) → (𝑢𝑀𝑣) = ((𝐹‘𝑢)𝑁(𝐹‘𝑣))))
25	24	imp 444	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝑋 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → (𝑢𝑀𝑣) = ((𝐹‘𝑢)𝑁(𝐹‘𝑣)))
26	25	an32s 842	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢𝑀𝑣) = ((𝐹‘𝑢)𝑁(𝐹‘𝑣)))
27	26	adantlrl 752	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝑋 ∧ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)))) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢𝑀𝑣) = ((𝐹‘𝑢)𝑁(𝐹‘𝑣)))
28	27	adantlll 750	. . . . . 6 ⊢ (((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ 𝐴 ⊆ 𝑋) ∧ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)))) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢𝑀𝑣) = ((𝐹‘𝑢)𝑁(𝐹‘𝑣)))
29		ismtyres.3	. . . . . . . . 9 ⊢ 𝑆 = (𝑀 ↾ (𝐴 × 𝐴))
30	29	oveqi 6562	. . . . . . . 8 ⊢ (𝑢𝑆𝑣) = (𝑢(𝑀 ↾ (𝐴 × 𝐴))𝑣)
31		ovres 6698	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢(𝑀 ↾ (𝐴 × 𝐴))𝑣) = (𝑢𝑀𝑣))
32	30, 31	syl5eq 2656	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢𝑆𝑣) = (𝑢𝑀𝑣))
33	32	adantl 481	. . . . . 6 ⊢ (((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ 𝐴 ⊆ 𝑋) ∧ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)))) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢𝑆𝑣) = (𝑢𝑀𝑣))
34		fvres 6117	. . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑢) = (𝐹‘𝑢))
35	34	ad2antrl 760	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝑢) = (𝐹‘𝑢))
36		fvres 6117	. . . . . . . . . . 11 ⊢ (𝑣 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑣) = (𝐹‘𝑣))
37	36	ad2antll 761	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝑣) = (𝐹‘𝑣))
38	35, 37	oveq12d 6567	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (((𝐹 ↾ 𝐴)‘𝑢)𝑇((𝐹 ↾ 𝐴)‘𝑣)) = ((𝐹‘𝑢)𝑇(𝐹‘𝑣)))
39		ismtyres.4	. . . . . . . . . . 11 ⊢ 𝑇 = (𝑁 ↾ (𝐵 × 𝐵))
40	39	oveqi 6562	. . . . . . . . . 10 ⊢ ((𝐹‘𝑢)𝑇(𝐹‘𝑣)) = ((𝐹‘𝑢)(𝑁 ↾ (𝐵 × 𝐵))(𝐹‘𝑣))
41		f1ofun 6052	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → Fun 𝐹)
42	41	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) → Fun 𝐹)
43		f1odm 6054	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → dom 𝐹 = 𝑋)
44	43	sseq2d 3596	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → (𝐴 ⊆ dom 𝐹 ↔ 𝐴 ⊆ 𝑋))
45	44	biimparc 503	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) → 𝐴 ⊆ dom 𝐹)
46		funfvima2 6397	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑢 ∈ 𝐴 → (𝐹‘𝑢) ∈ (𝐹 “ 𝐴)))
47	42, 45, 46	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑢 ∈ 𝐴 → (𝐹‘𝑢) ∈ (𝐹 “ 𝐴)))
48	47	imp 444	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑢 ∈ 𝐴) → (𝐹‘𝑢) ∈ (𝐹 “ 𝐴))
49		ismtyres.2	. . . . . . . . . . . . 13 ⊢ 𝐵 = (𝐹 “ 𝐴)
50	48, 49	syl6eleqr 2699	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑢 ∈ 𝐴) → (𝐹‘𝑢) ∈ 𝐵)
51	50	adantrr 749	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝐹‘𝑢) ∈ 𝐵)
52		funfvima2 6397	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑣 ∈ 𝐴 → (𝐹‘𝑣) ∈ (𝐹 “ 𝐴)))
53	42, 45, 52	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑣 ∈ 𝐴 → (𝐹‘𝑣) ∈ (𝐹 “ 𝐴)))
54	53	imp 444	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ (𝐹 “ 𝐴))
55	54, 49	syl6eleqr 2699	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ 𝐵)
56	55	adantrl 748	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝐹‘𝑣) ∈ 𝐵)
57	51, 56	ovresd 6699	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝐹‘𝑢)(𝑁 ↾ (𝐵 × 𝐵))(𝐹‘𝑣)) = ((𝐹‘𝑢)𝑁(𝐹‘𝑣)))
58	40, 57	syl5eq 2656	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝐹‘𝑢)𝑇(𝐹‘𝑣)) = ((𝐹‘𝑢)𝑁(𝐹‘𝑣)))
59	38, 58	eqtrd 2644	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (((𝐹 ↾ 𝐴)‘𝑢)𝑇((𝐹 ↾ 𝐴)‘𝑣)) = ((𝐹‘𝑢)𝑁(𝐹‘𝑣)))
60	59	adantlrr 753	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝑋 ∧ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)))) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (((𝐹 ↾ 𝐴)‘𝑢)𝑇((𝐹 ↾ 𝐴)‘𝑣)) = ((𝐹‘𝑢)𝑁(𝐹‘𝑣)))
61	60	adantlll 750	. . . . . 6 ⊢ (((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ 𝐴 ⊆ 𝑋) ∧ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)))) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (((𝐹 ↾ 𝐴)‘𝑢)𝑇((𝐹 ↾ 𝐴)‘𝑣)) = ((𝐹‘𝑢)𝑁(𝐹‘𝑣)))
62	28, 33, 61	3eqtr4d 2654	. . . . 5 ⊢ (((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ 𝐴 ⊆ 𝑋) ∧ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)))) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢𝑆𝑣) = (((𝐹 ↾ 𝐴)‘𝑢)𝑇((𝐹 ↾ 𝐴)‘𝑣)))
63	62	ralrimivva 2954	. . . 4 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ 𝐴 ⊆ 𝑋) ∧ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)))) → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢𝑆𝑣) = (((𝐹 ↾ 𝐴)‘𝑢)𝑇((𝐹 ↾ 𝐴)‘𝑣)))
64	63	adantlrl 752	. . 3 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) ∧ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)))) → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢𝑆𝑣) = (((𝐹 ↾ 𝐴)‘𝑢)𝑇((𝐹 ↾ 𝐴)‘𝑣)))
65	10, 64	mpdan 699	. 2 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢𝑆𝑣) = (((𝐹 ↾ 𝐴)‘𝑢)𝑇((𝐹 ↾ 𝐴)‘𝑣)))
66		xmetres2 21976	. . . . 5 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑀 ↾ (𝐴 × 𝐴)) ∈ (∞Met‘𝐴))
67	29, 66	syl5eqel 2692	. . . 4 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑆 ∈ (∞Met‘𝐴))
68	67	ad2ant2rl 781	. . 3 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝑆 ∈ (∞Met‘𝐴))
69		simplr 788	. . . . . 6 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝑁 ∈ (∞Met‘𝑌))
70		imassrn 5396	. . . . . . . 8 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
71	49, 70	eqsstri 3598	. . . . . . 7 ⊢ 𝐵 ⊆ ran 𝐹
72		f1ofo 6057	. . . . . . . 8 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
73		forn 6031	. . . . . . . 8 ⊢ (𝐹:𝑋–onto→𝑌 → ran 𝐹 = 𝑌)
74	3, 72, 73	3syl 18	. . . . . . 7 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → ran 𝐹 = 𝑌)
75	71, 74	syl5sseq 3616	. . . . . 6 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝐵 ⊆ 𝑌)
76		xmetres2 21976	. . . . . 6 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝐵 ⊆ 𝑌) → (𝑁 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
77	69, 75, 76	syl2anc 691	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → (𝑁 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
78	39, 77	syl5eqel 2692	. . . 4 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝑇 ∈ (∞Met‘𝐵))
79	49	fveq2i 6106	. . . 4 ⊢ (∞Met‘𝐵) = (∞Met‘(𝐹 “ 𝐴))
80	78, 79	syl6eleq 2698	. . 3 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝑇 ∈ (∞Met‘(𝐹 “ 𝐴)))
81		isismty 32770	. . 3 ⊢ ((𝑆 ∈ (∞Met‘𝐴) ∧ 𝑇 ∈ (∞Met‘(𝐹 “ 𝐴))) → ((𝐹 ↾ 𝐴) ∈ (𝑆 Ismty 𝑇) ↔ ((𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴) ∧ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢𝑆𝑣) = (((𝐹 ↾ 𝐴)‘𝑢)𝑇((𝐹 ↾ 𝐴)‘𝑣)))))
82	68, 80, 81	syl2anc 691	. 2 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → ((𝐹 ↾ 𝐴) ∈ (𝑆 Ismty 𝑇) ↔ ((𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴) ∧ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢𝑆𝑣) = (((𝐹 ↾ 𝐴)‘𝑢)𝑇((𝐹 ↾ 𝐴)‘𝑣)))))
83	8, 65, 82	mpbir2and 959	1 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → (𝐹 ↾ 𝐴) ∈ (𝑆 Ismty 𝑇))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540   × cxp 5036  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Fun wfun 5798  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ∞Metcxmt 19552   Ismty cismty 32767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-xr 9957  df-xmet 19560  df-ismty 32768

This theorem is referenced by:  reheibor  32808