| Step | Hyp | Ref
| Expression |
| 1 | | simpl 482 |
. . . 4
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋)) |
| 2 | | fvexd 6921 |
. . . . . . 7
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (unifTop‘𝑈) ∈ V) |
| 3 | | elfvex 6944 |
. . . . . . . . 9
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V) |
| 4 | 3 | adantr 480 |
. . . . . . . 8
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ V) |
| 5 | | simpr 484 |
. . . . . . . 8
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋) |
| 6 | 4, 5 | ssexd 5324 |
. . . . . . 7
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V) |
| 7 | | elrest 17472 |
. . . . . . 7
⊢
(((unifTop‘𝑈)
∈ V ∧ 𝐴 ∈ V)
→ (𝑏 ∈
((unifTop‘𝑈)
↾t 𝐴)
↔ ∃𝑎 ∈
(unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴))) |
| 8 | 2, 6, 7 | syl2anc 584 |
. . . . . 6
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) ↔ ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴))) |
| 9 | 8 | biimpa 476 |
. . . . 5
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴)) |
| 10 | | inss2 4238 |
. . . . . . 7
⊢ (𝑎 ∩ 𝐴) ⊆ 𝐴 |
| 11 | | sseq1 4009 |
. . . . . . 7
⊢ (𝑏 = (𝑎 ∩ 𝐴) → (𝑏 ⊆ 𝐴 ↔ (𝑎 ∩ 𝐴) ⊆ 𝐴)) |
| 12 | 10, 11 | mpbiri 258 |
. . . . . 6
⊢ (𝑏 = (𝑎 ∩ 𝐴) → 𝑏 ⊆ 𝐴) |
| 13 | 12 | rexlimivw 3151 |
. . . . 5
⊢
(∃𝑎 ∈
(unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴) → 𝑏 ⊆ 𝐴) |
| 14 | 9, 13 | syl 17 |
. . . 4
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → 𝑏 ⊆ 𝐴) |
| 15 | | simp-5l 785 |
. . . . . . . . . 10
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) → 𝑈 ∈ (UnifOn‘𝑋)) |
| 16 | 15 | ad2antrr 726 |
. . . . . . . . 9
⊢
((((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝑈 ∈ (UnifOn‘𝑋)) |
| 17 | 6 | ad6antr 736 |
. . . . . . . . . 10
⊢
((((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝐴 ∈ V) |
| 18 | 17, 17 | xpexd 7771 |
. . . . . . . . 9
⊢
((((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → (𝐴 × 𝐴) ∈ V) |
| 19 | | simplr 769 |
. . . . . . . . 9
⊢
((((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝑢 ∈ 𝑈) |
| 20 | | elrestr 17473 |
. . . . . . . . 9
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V ∧ 𝑢 ∈ 𝑈) → (𝑢 ∩ (𝐴 × 𝐴)) ∈ (𝑈 ↾t (𝐴 × 𝐴))) |
| 21 | 16, 18, 19, 20 | syl3anc 1373 |
. . . . . . . 8
⊢
((((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → (𝑢 ∩ (𝐴 × 𝐴)) ∈ (𝑈 ↾t (𝐴 × 𝐴))) |
| 22 | | inss1 4237 |
. . . . . . . . . . . . 13
⊢ (𝑢 ∩ (𝐴 × 𝐴)) ⊆ 𝑢 |
| 23 | | imass1 6119 |
. . . . . . . . . . . . 13
⊢ ((𝑢 ∩ (𝐴 × 𝐴)) ⊆ 𝑢 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑢 “ {𝑥})) |
| 24 | 22, 23 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑢 “ {𝑥}) |
| 25 | | sstr 3992 |
. . . . . . . . . . . 12
⊢ ((((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑢 “ {𝑥}) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑎) |
| 26 | 24, 25 | mpan 690 |
. . . . . . . . . . 11
⊢ ((𝑢 “ {𝑥}) ⊆ 𝑎 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑎) |
| 27 | | imassrn 6089 |
. . . . . . . . . . . . . . 15
⊢ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ ran (𝑢 ∩ (𝐴 × 𝐴)) |
| 28 | | rnin 6166 |
. . . . . . . . . . . . . . 15
⊢ ran
(𝑢 ∩ (𝐴 × 𝐴)) ⊆ (ran 𝑢 ∩ ran (𝐴 × 𝐴)) |
| 29 | 27, 28 | sstri 3993 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (ran 𝑢 ∩ ran (𝐴 × 𝐴)) |
| 30 | | inss2 4238 |
. . . . . . . . . . . . . 14
⊢ (ran
𝑢 ∩ ran (𝐴 × 𝐴)) ⊆ ran (𝐴 × 𝐴) |
| 31 | 29, 30 | sstri 3993 |
. . . . . . . . . . . . 13
⊢ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ ran (𝐴 × 𝐴) |
| 32 | | rnxpid 6193 |
. . . . . . . . . . . . 13
⊢ ran
(𝐴 × 𝐴) = 𝐴 |
| 33 | 31, 32 | sseqtri 4032 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝐴 |
| 34 | 33 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝑢 “ {𝑥}) ⊆ 𝑎 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝐴) |
| 35 | 26, 34 | ssind 4241 |
. . . . . . . . . 10
⊢ ((𝑢 “ {𝑥}) ⊆ 𝑎 → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑎 ∩ 𝐴)) |
| 36 | 35 | adantl 481 |
. . . . . . . . 9
⊢
((((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ (𝑎 ∩ 𝐴)) |
| 37 | | simpllr 776 |
. . . . . . . . 9
⊢
((((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → 𝑏 = (𝑎 ∩ 𝐴)) |
| 38 | 36, 37 | sseqtrrd 4021 |
. . . . . . . 8
⊢
((((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑏) |
| 39 | | imaeq1 6073 |
. . . . . . . . . 10
⊢ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) → (𝑣 “ {𝑥}) = ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥})) |
| 40 | 39 | sseq1d 4015 |
. . . . . . . . 9
⊢ (𝑣 = (𝑢 ∩ (𝐴 × 𝐴)) → ((𝑣 “ {𝑥}) ⊆ 𝑏 ↔ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑏)) |
| 41 | 40 | rspcev 3622 |
. . . . . . . 8
⊢ (((𝑢 ∩ (𝐴 × 𝐴)) ∈ (𝑈 ↾t (𝐴 × 𝐴)) ∧ ((𝑢 ∩ (𝐴 × 𝐴)) “ {𝑥}) ⊆ 𝑏) → ∃𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏) |
| 42 | 21, 38, 41 | syl2anc 584 |
. . . . . . 7
⊢
((((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 “ {𝑥}) ⊆ 𝑎) → ∃𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏) |
| 43 | | simplr 769 |
. . . . . . . 8
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) → 𝑎 ∈ (unifTop‘𝑈)) |
| 44 | | simpllr 776 |
. . . . . . . . . 10
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) → 𝑥 ∈ 𝑏) |
| 45 | | simpr 484 |
. . . . . . . . . 10
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) → 𝑏 = (𝑎 ∩ 𝐴)) |
| 46 | 44, 45 | eleqtrd 2843 |
. . . . . . . . 9
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) → 𝑥 ∈ (𝑎 ∩ 𝐴)) |
| 47 | 46 | elin1d 4204 |
. . . . . . . 8
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) → 𝑥 ∈ 𝑎) |
| 48 | | elutop 24242 |
. . . . . . . . . 10
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑎 ∈ (unifTop‘𝑈) ↔ (𝑎 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑎 ∃𝑢 ∈ 𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎))) |
| 49 | 48 | simplbda 499 |
. . . . . . . . 9
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ (unifTop‘𝑈)) → ∀𝑥 ∈ 𝑎 ∃𝑢 ∈ 𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎) |
| 50 | 49 | r19.21bi 3251 |
. . . . . . . 8
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑥 ∈ 𝑎) → ∃𝑢 ∈ 𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎) |
| 51 | 15, 43, 47, 50 | syl21anc 838 |
. . . . . . 7
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) → ∃𝑢 ∈ 𝑈 (𝑢 “ {𝑥}) ⊆ 𝑎) |
| 52 | 42, 51 | r19.29a 3162 |
. . . . . 6
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) ∧ 𝑎 ∈ (unifTop‘𝑈)) ∧ 𝑏 = (𝑎 ∩ 𝐴)) → ∃𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏) |
| 53 | 9 | adantr 480 |
. . . . . 6
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) → ∃𝑎 ∈ (unifTop‘𝑈)𝑏 = (𝑎 ∩ 𝐴)) |
| 54 | 52, 53 | r19.29a 3162 |
. . . . 5
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) ∧ 𝑥 ∈ 𝑏) → ∃𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏) |
| 55 | 54 | ralrimiva 3146 |
. . . 4
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → ∀𝑥 ∈ 𝑏 ∃𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏) |
| 56 | | trust 24238 |
. . . . . 6
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴)) |
| 57 | | elutop 24242 |
. . . . . 6
⊢ ((𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) → (𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))) ↔ (𝑏 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑏 ∃𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏))) |
| 58 | 56, 57 | syl 17 |
. . . . 5
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))) ↔ (𝑏 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑏 ∃𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏))) |
| 59 | 58 | biimpar 477 |
. . . 4
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ (𝑏 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑏 ∃𝑣 ∈ (𝑈 ↾t (𝐴 × 𝐴))(𝑣 “ {𝑥}) ⊆ 𝑏)) → 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) |
| 60 | 1, 14, 55, 59 | syl12anc 837 |
. . 3
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴)) → 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) |
| 61 | 60 | ex 412 |
. 2
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑏 ∈ ((unifTop‘𝑈) ↾t 𝐴) → 𝑏 ∈ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))))) |
| 62 | 61 | ssrdv 3989 |
1
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴)))) |