Step | Hyp | Ref
| Expression |
1 | | isismty 35696 |
. . . . . 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 502 |
. . . . 5
⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → 𝐹:𝑋–1-1-onto→𝑌) |
3 | 2 | adantrr 717 |
. . . 4
⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝐹:𝑋–1-1-onto→𝑌) |
4 | | f1of1 6660 |
. . . 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 773 |
. . 3
⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝐴 ⊆ 𝑋) |
7 | | f1ores 6675 |
. . 3
⊢ ((𝐹:𝑋–1-1→𝑌 ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴)) |
8 | 5, 6, 7 | syl2anc 587 |
. 2
⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴)) |
9 | 1 | biimpa 480 |
. . . 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 717 |
. . 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 3893 |
. . . . . . . . . . . . 13
⊢ (𝐴 ⊆ 𝑋 → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝑋)) |
12 | | ssel 3893 |
. . . . . . . . . . . . 13
⊢ (𝐴 ⊆ 𝑋 → (𝑣 ∈ 𝐴 → 𝑣 ∈ 𝑋)) |
13 | 11, 12 | anim12d 612 |
. . . . . . . . . . . 12
⊢ (𝐴 ⊆ 𝑋 → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋))) |
14 | 13 | imp 410 |
. . . . . . . . . . 11
⊢ ((𝐴 ⊆ 𝑋 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)) |
15 | | oveq1 7220 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑢 → (𝑥𝑀𝑦) = (𝑢𝑀𝑦)) |
16 | | fveq2 6717 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑢 → (𝐹‘𝑥) = (𝐹‘𝑢)) |
17 | 16 | oveq1d 7228 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑢 → ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) = ((𝐹‘𝑢)𝑁(𝐹‘𝑦))) |
18 | 15, 17 | eqeq12d 2753 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑢 → ((𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) ↔ (𝑢𝑀𝑦) = ((𝐹‘𝑢)𝑁(𝐹‘𝑦)))) |
19 | | oveq2 7221 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑣 → (𝑢𝑀𝑦) = (𝑢𝑀𝑣)) |
20 | | fveq2 6717 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑣 → (𝐹‘𝑦) = (𝐹‘𝑣)) |
21 | 20 | oveq2d 7229 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑣 → ((𝐹‘𝑢)𝑁(𝐹‘𝑦)) = ((𝐹‘𝑢)𝑁(𝐹‘𝑣))) |
22 | 19, 21 | eqeq12d 2753 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑣 → ((𝑢𝑀𝑦) = ((𝐹‘𝑢)𝑁(𝐹‘𝑦)) ↔ (𝑢𝑀𝑣) = ((𝐹‘𝑢)𝑁(𝐹‘𝑣)))) |
23 | 18, 22 | rspc2v 3547 |
. . . . . . . . . . 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 410 |
. . . . . . . . 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 652 |
. . . . . . . 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 720 |
. . . . . . 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 718 |
. . . . . 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 7226 |
. . . . . . . 8
⊢ (𝑢𝑆𝑣) = (𝑢(𝑀 ↾ (𝐴 × 𝐴))𝑣) |
31 | | ovres 7374 |
. . . . . . . 8
⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢(𝑀 ↾ (𝐴 × 𝐴))𝑣) = (𝑢𝑀𝑣)) |
32 | 30, 31 | syl5eq 2790 |
. . . . . . 7
⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢𝑆𝑣) = (𝑢𝑀𝑣)) |
33 | 32 | adantl 485 |
. . . . . 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 6736 |
. . . . . . . . . . 11
⊢ (𝑢 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑢) = (𝐹‘𝑢)) |
35 | 34 | ad2antrl 728 |
. . . . . . . . . 10
⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝑢) = (𝐹‘𝑢)) |
36 | | fvres 6736 |
. . . . . . . . . . 11
⊢ (𝑣 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑣) = (𝐹‘𝑣)) |
37 | 36 | ad2antll 729 |
. . . . . . . . . 10
⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝑣) = (𝐹‘𝑣)) |
38 | 35, 37 | oveq12d 7231 |
. . . . . . . . 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 7226 |
. . . . . . . . . 10
⊢ ((𝐹‘𝑢)𝑇(𝐹‘𝑣)) = ((𝐹‘𝑢)(𝑁 ↾ (𝐵 × 𝐵))(𝐹‘𝑣)) |
41 | | f1ofun 6663 |
. . . . . . . . . . . . . . . 16
⊢ (𝐹:𝑋–1-1-onto→𝑌 → Fun 𝐹) |
42 | 41 | adantl 485 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) → Fun 𝐹) |
43 | | f1odm 6665 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐹:𝑋–1-1-onto→𝑌 → dom 𝐹 = 𝑋) |
44 | 43 | sseq2d 3933 |
. . . . . . . . . . . . . . . 16
⊢ (𝐹:𝑋–1-1-onto→𝑌 → (𝐴 ⊆ dom 𝐹 ↔ 𝐴 ⊆ 𝑋)) |
45 | 44 | biimparc 483 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) → 𝐴 ⊆ dom 𝐹) |
46 | | funfvima2 7047 |
. . . . . . . . . . . . . . 15
⊢ ((Fun
𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑢 ∈ 𝐴 → (𝐹‘𝑢) ∈ (𝐹 “ 𝐴))) |
47 | 42, 45, 46 | syl2anc 587 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑢 ∈ 𝐴 → (𝐹‘𝑢) ∈ (𝐹 “ 𝐴))) |
48 | 47 | imp 410 |
. . . . . . . . . . . . 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 2849 |
. . . . . . . . . . . 12
⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑢 ∈ 𝐴) → (𝐹‘𝑢) ∈ 𝐵) |
51 | 50 | adantrr 717 |
. . . . . . . . . . 11
⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝐹‘𝑢) ∈ 𝐵) |
52 | | funfvima2 7047 |
. . . . . . . . . . . . . . 15
⊢ ((Fun
𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑣 ∈ 𝐴 → (𝐹‘𝑣) ∈ (𝐹 “ 𝐴))) |
53 | 42, 45, 52 | syl2anc 587 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑣 ∈ 𝐴 → (𝐹‘𝑣) ∈ (𝐹 “ 𝐴))) |
54 | 53 | imp 410 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ (𝐹 “ 𝐴)) |
55 | 54, 49 | eleqtrrdi 2849 |
. . . . . . . . . . . 12
⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ 𝐵) |
56 | 55 | adantrl 716 |
. . . . . . . . . . 11
⊢ (((𝐴 ⊆ 𝑋 ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝐹‘𝑣) ∈ 𝐵) |
57 | 51, 56 | ovresd 7375 |
. . . . . . . . . 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 2790 |
. . . . . . . . 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 2777 |
. . . . . . . 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 721 |
. . . . . . 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 718 |
. . . . . 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 2787 |
. . . . 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 3112 |
. . . 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 720 |
. . 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 687 |
. 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 23259 |
. . . . 5
⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑀 ↾ (𝐴 × 𝐴)) ∈ (∞Met‘𝐴)) |
67 | 29, 66 | eqeltrid 2842 |
. . . 4
⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑆 ∈ (∞Met‘𝐴)) |
68 | 67 | ad2ant2rl 749 |
. . 3
⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝑆 ∈ (∞Met‘𝐴)) |
69 | | simplr 769 |
. . . . . 6
⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝑁 ∈ (∞Met‘𝑌)) |
70 | | imassrn 5940 |
. . . . . . . 8
⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹 |
71 | 49, 70 | eqsstri 3935 |
. . . . . . 7
⊢ 𝐵 ⊆ ran 𝐹 |
72 | | f1ofo 6668 |
. . . . . . . 8
⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌) |
73 | | forn 6636 |
. . . . . . . 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 | sseqtrid 3953 |
. . . . . 6
⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝐵 ⊆ 𝑌) |
76 | | xmetres2 23259 |
. . . . . 6
⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝐵 ⊆ 𝑌) → (𝑁 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵)) |
77 | 69, 75, 76 | syl2anc 587 |
. . . . 5
⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → (𝑁 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵)) |
78 | 39, 77 | eqeltrid 2842 |
. . . 4
⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝑇 ∈ (∞Met‘𝐵)) |
79 | 49 | fveq2i 6720 |
. . . 4
⊢
(∞Met‘𝐵)
= (∞Met‘(𝐹
“ 𝐴)) |
80 | 78, 79 | eleqtrdi 2848 |
. . 3
⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → 𝑇 ∈ (∞Met‘(𝐹 “ 𝐴))) |
81 | | isismty 35696 |
. . 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 587 |
. 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 713 |
1
⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → (𝐹 ↾ 𝐴) ∈ (𝑆 Ismty 𝑇)) |