Step | Hyp | Ref
| Expression |
1 | | ixpfn 8450 |
. . . . . 6
⊢ (𝑓 ∈ X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) → 𝑓 Fn 𝑋) |
2 | 1 | adantl 485 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) → 𝑓 Fn 𝑋) |
3 | | rrxsnicc.1 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m 𝑋)) |
4 | | elmapfn 8412 |
. . . . . . 7
⊢ (𝐴 ∈ (ℝ
↑m 𝑋)
→ 𝐴 Fn 𝑋) |
5 | 3, 4 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐴 Fn 𝑋) |
6 | 5 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) → 𝐴 Fn 𝑋) |
7 | | simpll 766 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) ∧ 𝑗 ∈ 𝑋) → 𝜑) |
8 | | fveq2 6645 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑗 → (𝐴‘𝑘) = (𝐴‘𝑗)) |
9 | 8, 8 | oveq12d 7153 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘)[,](𝐴‘𝑘)) = ((𝐴‘𝑗)[,](𝐴‘𝑗))) |
10 | 9 | cbvixpv 8462 |
. . . . . . . . 9
⊢ X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗)) |
11 | 10 | eleq2i 2881 |
. . . . . . . 8
⊢ (𝑓 ∈ X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) ↔ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) |
12 | 11 | biimpi 219 |
. . . . . . 7
⊢ (𝑓 ∈ X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) |
13 | 12 | ad2antlr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) ∧ 𝑗 ∈ 𝑋) → 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) |
14 | | simpr 488 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) ∧ 𝑗 ∈ 𝑋) → 𝑗 ∈ 𝑋) |
15 | | elmapi 8411 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ (ℝ
↑m 𝑋)
→ 𝐴:𝑋⟶ℝ) |
16 | 3, 15 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴:𝑋⟶ℝ) |
17 | 16 | ffvelrnda 6828 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ∈ ℝ) |
18 | 17 | adantlr 714 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ∈ ℝ) |
19 | 18, 18 | iccssred 12812 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → ((𝐴‘𝑗)[,](𝐴‘𝑗)) ⊆ ℝ) |
20 | | fvixp2 41827 |
. . . . . . . . . 10
⊢ ((𝑓 ∈ X𝑗 ∈
𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗)) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ∈ ((𝐴‘𝑗)[,](𝐴‘𝑗))) |
21 | 20 | adantll 713 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ∈ ((𝐴‘𝑗)[,](𝐴‘𝑗))) |
22 | 19, 21 | sseldd 3916 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ∈ ℝ) |
23 | 22 | rexrd 10680 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ∈
ℝ*) |
24 | 18 | rexrd 10680 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ∈
ℝ*) |
25 | | iccleub 12780 |
. . . . . . . 8
⊢ (((𝐴‘𝑗) ∈ ℝ* ∧ (𝐴‘𝑗) ∈ ℝ* ∧ (𝑓‘𝑗) ∈ ((𝐴‘𝑗)[,](𝐴‘𝑗))) → (𝑓‘𝑗) ≤ (𝐴‘𝑗)) |
26 | 24, 24, 21, 25 | syl3anc 1368 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) ≤ (𝐴‘𝑗)) |
27 | | iccgelb 12781 |
. . . . . . . 8
⊢ (((𝐴‘𝑗) ∈ ℝ* ∧ (𝐴‘𝑗) ∈ ℝ* ∧ (𝑓‘𝑗) ∈ ((𝐴‘𝑗)[,](𝐴‘𝑗))) → (𝐴‘𝑗) ≤ (𝑓‘𝑗)) |
28 | 24, 24, 21, 27 | syl3anc 1368 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ≤ (𝑓‘𝑗)) |
29 | 23, 24, 26, 28 | xrletrid 12536 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑗 ∈ 𝑋 ((𝐴‘𝑗)[,](𝐴‘𝑗))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) = (𝐴‘𝑗)) |
30 | 7, 13, 14, 29 | syl21anc 836 |
. . . . 5
⊢ (((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) ∧ 𝑗 ∈ 𝑋) → (𝑓‘𝑗) = (𝐴‘𝑗)) |
31 | 2, 6, 30 | eqfnfvd 6782 |
. . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) → 𝑓 = 𝐴) |
32 | | velsn 4541 |
. . . . . 6
⊢ (𝑓 ∈ {𝐴} ↔ 𝑓 = 𝐴) |
33 | 32 | bicomi 227 |
. . . . 5
⊢ (𝑓 = 𝐴 ↔ 𝑓 ∈ {𝐴}) |
34 | 33 | biimpi 219 |
. . . 4
⊢ (𝑓 = 𝐴 → 𝑓 ∈ {𝐴}) |
35 | 31, 34 | syl 17 |
. . 3
⊢ ((𝜑 ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) → 𝑓 ∈ {𝐴}) |
36 | 35 | ssd 41716 |
. 2
⊢ (𝜑 → X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) ⊆ {𝐴}) |
37 | 3 | elexd 3461 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ V) |
38 | 16 | ffvelrnda 6828 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) |
39 | 38 | leidd 11195 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (𝐴‘𝑘)) |
40 | 38, 38, 38, 39, 39 | eliccd 42141 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ((𝐴‘𝑘)[,](𝐴‘𝑘))) |
41 | 40 | ralrimiva 3149 |
. . . . 5
⊢ (𝜑 → ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) ∈ ((𝐴‘𝑘)[,](𝐴‘𝑘))) |
42 | 37, 5, 41 | 3jca 1125 |
. . . 4
⊢ (𝜑 → (𝐴 ∈ V ∧ 𝐴 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) ∈ ((𝐴‘𝑘)[,](𝐴‘𝑘)))) |
43 | | elixp2 8448 |
. . . 4
⊢ (𝐴 ∈ X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) ↔ (𝐴 ∈ V ∧ 𝐴 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) ∈ ((𝐴‘𝑘)[,](𝐴‘𝑘)))) |
44 | 42, 43 | sylibr 237 |
. . 3
⊢ (𝜑 → 𝐴 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) |
45 | | snssg 4678 |
. . . 4
⊢ (𝐴 ∈ (ℝ
↑m 𝑋)
→ (𝐴 ∈ X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) ↔ {𝐴} ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)))) |
46 | 3, 45 | syl 17 |
. . 3
⊢ (𝜑 → (𝐴 ∈ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) ↔ {𝐴} ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)))) |
47 | 44, 46 | mpbid 235 |
. 2
⊢ (𝜑 → {𝐴} ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘))) |
48 | 36, 47 | eqssd 3932 |
1
⊢ (𝜑 → X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {𝐴}) |