Theorem ismtyres 38381

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 38374	. . . . . 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 503	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → 𝐹:𝑋–1-1-onto→𝑌)
3	2	adantrr 729	. . . 4 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝐹:𝑋–1-1-onto→𝑌)
4		f1of1 6820	. . . 4 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–1-1→𝑌)
5	3, 4	syl 18	. . 3 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝐹:𝑋–1-1→𝑌)
6		simprr 784	. . 3 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝐴 ⊆ 𝑋)
7		f1ores 6836	. . 3 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴))
8	5, 6, 7	syl2anc 595	. 2 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴))
9	1	biimpa 481	. . . 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 729	. . 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 3939	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑋 → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑋))
12		ssel 3939	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑋 → (𝑣 ∈ 𝐴 → 𝑣 ∈ 𝑋))
13	11, 12	anim12d 620	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝑋 → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)))
14	13	imp 411	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝑋 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋))
15		oveq1 7418	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → (𝑥𝑀𝑦) = (𝑢𝑀𝑦))
16		fveq2 6882	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑢 → (𝐹‘𝑥) = (𝐹‘𝑢))
17	16	oveq1d 7426	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑢 → ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) = ((𝐹‘𝑢)𝑁(𝐹‘𝑦)))
18	15, 17	eqeq12d 2785	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → ((𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) ↔ (𝑢𝑀𝑦) = ((𝐹‘𝑢)𝑁(𝐹‘𝑦))))
19		oveq2 7419	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑣 → (𝑢𝑀𝑦) = (𝑢𝑀𝑣))
20		fveq2 6882	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑣 → (𝐹‘𝑦) = (𝐹‘𝑣))
21	20	oveq2d 7427	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑣 → ((𝐹‘𝑢)𝑁(𝐹‘𝑦)) = ((𝐹‘𝑢)𝑁(𝐹‘𝑣)))
22	19, 21	eqeq12d 2785	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑣 → ((𝑢𝑀𝑦) = ((𝐹‘𝑢)𝑁(𝐹‘𝑦)) ↔ (𝑢𝑀𝑣) = ((𝐹‘𝑢)𝑁(𝐹‘𝑣))))
23	18, 22	rspc2v 3601	. . . . . . . . . . 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 18	. . . . . . . . . 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 411	. . . . . . . . 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 664	. . . . . . . 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 732	. . . . . . 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 730	. . . . . 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 7424	. . . . . . . 8 ⊢ (𝑢𝑆𝑣) = (𝑢(𝑀 ↾ (𝐴 × 𝐴))𝑣)
31		ovres 7577	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢(𝑀 ↾ (𝐴 × 𝐴))𝑣) = (𝑢𝑀𝑣))
32	30, 31	eqtrid 2816	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢𝑆𝑣) = (𝑢𝑀𝑣))
33	32	adantl 486	. . . . . 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 6901	. . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑢) = (𝐹‘𝑢))
35	34	ad2antrl 740	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝑢) = (𝐹‘𝑢))
36		fvres 6901	. . . . . . . . . . 11 ⊢ (𝑣 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑣) = (𝐹‘𝑣))
37	36	ad2antll 741	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝑣) = (𝐹‘𝑣))
38	35, 37	oveq12d 7429	. . . . . . . . 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 7424	. . . . . . . . . 10 ⊢ ((𝐹‘𝑢)𝑇(𝐹‘𝑣)) = ((𝐹‘𝑢)(𝑁 ↾ (𝐵 × 𝐵))(𝐹‘𝑣))
41		f1ofun 6823	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → Fun 𝐹)
42	41	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) → Fun 𝐹)
43		f1odm 6825	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → dom 𝐹 = 𝑋)
44	43	sseq2d 3977	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → (𝐴 ⊆ dom 𝐹 ↔ 𝐴 ⊆ 𝑋))
45	44	biimparc 484	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) → 𝐴 ⊆ dom 𝐹)
46		funfvima2 7230	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑢 ∈ 𝐴 → (𝐹‘𝑢) ∈ (𝐹 “ 𝐴)))
47	42, 45, 46	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑢 ∈ 𝐴 → (𝐹‘𝑢) ∈ (𝐹 “ 𝐴)))
48	47	imp 411	. . . . . . . . . . . . 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	eleqtrrdi 2880	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑢 ∈ 𝐴) → (𝐹‘𝑢) ∈ 𝐵)
51	50	adantrr 729	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝐹‘𝑢) ∈ 𝐵)
52		funfvima2 7230	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑣 ∈ 𝐴 → (𝐹‘𝑣) ∈ (𝐹 “ 𝐴)))
53	42, 45, 52	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑣 ∈ 𝐴 → (𝐹‘𝑣) ∈ (𝐹 “ 𝐴)))
54	53	imp 411	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ (𝐹 “ 𝐴))
55	54, 49	eleqtrrdi 2880	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ 𝐵)
56	55	adantrl 728	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝐹‘𝑣) ∈ 𝐵)
57	51, 56	ovresd 7578	. . . . . . . . . 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	eqtrid 2816	. . . . . . . . 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 2804	. . . . . . . 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 733	. . . . . . 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 730	. . . . . 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 2814	. . . . 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 3214	. . . 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 732	. . 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 24487	. . . . 5 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑀 ↾ (𝐴 × 𝐴)) ∈ (∞Met‘𝐴))
67	29, 66	eqeltrid 2873	. . . 4 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑆 ∈ (∞Met‘𝐴))
68	67	ad2ant2rl 761	. . 3 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝑆 ∈ (∞Met‘𝐴))
69		simplr 780	. . . . . 6 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝑁 ∈ (∞Met‘𝑌))
70		imassrn 6074	. . . . . . . 8 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
71	49, 70	eqsstri 3991	. . . . . . 7 ⊢ 𝐵 ⊆ ran 𝐹
72		f1ofo 6829	. . . . . . . 8 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
73		forn 6796	. . . . . . . 8 ⊢ (𝐹:𝑋–onto→𝑌 → ran 𝐹 = 𝑌)
74	3, 72, 73	3syl 19	. . . . . . 7 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → ran 𝐹 = 𝑌)
75	71, 74	sseqtrid 3987	. . . . . 6 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝐵 ⊆ 𝑌)
76		xmetres2 24487	. . . . . 6 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝐵 ⊆ 𝑌) → (𝑁 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
77	69, 75, 76	syl2anc 595	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → (𝑁 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
78	39, 77	eqeltrid 2873	. . . 4 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝑇 ∈ (∞Met‘𝐵))
79	49	fveq2i 6885	. . . 4 ⊢ (∞Met‘𝐵) = (∞Met‘(𝐹 “ 𝐴))
80	78, 79	eleqtrdi 2879	. . 3 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝑇 ∈ (∞Met‘(𝐹 “ 𝐴)))
81		isismty 38374	. . 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 595	. 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 725	1 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → (𝐹 ↾ 𝐴) ∈ (𝑆 Ismty 𝑇))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085   ⊆ wss 3913   × cxp 5660  dom cdm 5662  ran crn 5663   ↾ cres 5664   “ cima 5665  Fun wfun 6531  –1-1→wf1 6534  –onto→wfo 6535  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411  ∞Metcxmet 21476   Ismty cismty 38371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8826  df-xr 11247  df-xmet 21484  df-ismty 38372

This theorem is referenced by:  reheibor  38412