Step |
Hyp |
Ref |
Expression |
1 |
|
issmfgt.s |
⊢ ( 𝜑 → 𝑆 ∈ SAlg ) |
2 |
|
issmfgt.d |
⊢ 𝐷 = dom 𝐹 |
3 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → 𝑆 ∈ SAlg ) |
4 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) |
5 |
3 4 2
|
smfdmss |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → 𝐷 ⊆ ∪ 𝑆 ) |
6 |
3 4 2
|
smff |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → 𝐹 : 𝐷 ⟶ ℝ ) |
7 |
|
nfv |
⊢ Ⅎ 𝑏 𝜑 |
8 |
|
nfv |
⊢ Ⅎ 𝑏 𝐹 ∈ ( SMblFn ‘ 𝑆 ) |
9 |
7 8
|
nfan |
⊢ Ⅎ 𝑏 ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) |
10 |
3 5
|
restuni4 |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ∪ ( 𝑆 ↾t 𝐷 ) = 𝐷 ) |
11 |
10
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → 𝐷 = ∪ ( 𝑆 ↾t 𝐷 ) ) |
12 |
11
|
rabeqdv |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } = { 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ) |
13 |
12
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑏 ∈ ℝ ) → { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } = { 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ) |
14 |
|
nfv |
⊢ Ⅎ 𝑦 𝜑 |
15 |
|
nfv |
⊢ Ⅎ 𝑦 𝐹 ∈ ( SMblFn ‘ 𝑆 ) |
16 |
14 15
|
nfan |
⊢ Ⅎ 𝑦 ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) |
17 |
|
nfv |
⊢ Ⅎ 𝑦 𝑏 ∈ ℝ |
18 |
16 17
|
nfan |
⊢ Ⅎ 𝑦 ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑏 ∈ ℝ ) |
19 |
|
nfv |
⊢ Ⅎ 𝑐 ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑏 ∈ ℝ ) |
20 |
1
|
uniexd |
⊢ ( 𝜑 → ∪ 𝑆 ∈ V ) |
21 |
20
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐷 ⊆ ∪ 𝑆 ) → ∪ 𝑆 ∈ V ) |
22 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐷 ⊆ ∪ 𝑆 ) → 𝐷 ⊆ ∪ 𝑆 ) |
23 |
21 22
|
ssexd |
⊢ ( ( 𝜑 ∧ 𝐷 ⊆ ∪ 𝑆 ) → 𝐷 ∈ V ) |
24 |
5 23
|
syldan |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → 𝐷 ∈ V ) |
25 |
|
eqid |
⊢ ( 𝑆 ↾t 𝐷 ) = ( 𝑆 ↾t 𝐷 ) |
26 |
3 24 25
|
subsalsal |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ( 𝑆 ↾t 𝐷 ) ∈ SAlg ) |
27 |
26
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑏 ∈ ℝ ) → ( 𝑆 ↾t 𝐷 ) ∈ SAlg ) |
28 |
|
eqid |
⊢ ∪ ( 𝑆 ↾t 𝐷 ) = ∪ ( 𝑆 ↾t 𝐷 ) |
29 |
6
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ) → 𝐹 : 𝐷 ⟶ ℝ ) |
30 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ) → 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ) |
31 |
10
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ) → ∪ ( 𝑆 ↾t 𝐷 ) = 𝐷 ) |
32 |
30 31
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ) → 𝑦 ∈ 𝐷 ) |
33 |
29 32
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ ℝ ) |
34 |
33
|
rexrd |
⊢ ( ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ ℝ* ) |
35 |
34
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑏 ∈ ℝ ) ∧ 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ ℝ* ) |
36 |
3 2
|
issmfle |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ( 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ↔ ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑐 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑦 ) ≤ 𝑐 } ∈ ( 𝑆 ↾t 𝐷 ) ) ) ) |
37 |
4 36
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑐 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑦 ) ≤ 𝑐 } ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
38 |
37
|
simp3d |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ∀ 𝑐 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑦 ) ≤ 𝑐 } ∈ ( 𝑆 ↾t 𝐷 ) ) |
39 |
10
|
rabeqdv |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → { 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ∣ ( 𝐹 ‘ 𝑦 ) ≤ 𝑐 } = { 𝑦 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑦 ) ≤ 𝑐 } ) |
40 |
39
|
eleq1d |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ( { 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ∣ ( 𝐹 ‘ 𝑦 ) ≤ 𝑐 } ∈ ( 𝑆 ↾t 𝐷 ) ↔ { 𝑦 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑦 ) ≤ 𝑐 } ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
41 |
40
|
ralbidv |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ( ∀ 𝑐 ∈ ℝ { 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ∣ ( 𝐹 ‘ 𝑦 ) ≤ 𝑐 } ∈ ( 𝑆 ↾t 𝐷 ) ↔ ∀ 𝑐 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑦 ) ≤ 𝑐 } ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
42 |
38 41
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ∀ 𝑐 ∈ ℝ { 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ∣ ( 𝐹 ‘ 𝑦 ) ≤ 𝑐 } ∈ ( 𝑆 ↾t 𝐷 ) ) |
43 |
42
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑐 ∈ ℝ ) → ∀ 𝑐 ∈ ℝ { 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ∣ ( 𝐹 ‘ 𝑦 ) ≤ 𝑐 } ∈ ( 𝑆 ↾t 𝐷 ) ) |
44 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑐 ∈ ℝ ) → 𝑐 ∈ ℝ ) |
45 |
|
rspa |
⊢ ( ( ∀ 𝑐 ∈ ℝ { 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ∣ ( 𝐹 ‘ 𝑦 ) ≤ 𝑐 } ∈ ( 𝑆 ↾t 𝐷 ) ∧ 𝑐 ∈ ℝ ) → { 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ∣ ( 𝐹 ‘ 𝑦 ) ≤ 𝑐 } ∈ ( 𝑆 ↾t 𝐷 ) ) |
46 |
43 44 45
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑐 ∈ ℝ ) → { 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ∣ ( 𝐹 ‘ 𝑦 ) ≤ 𝑐 } ∈ ( 𝑆 ↾t 𝐷 ) ) |
47 |
46
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑏 ∈ ℝ ) ∧ 𝑐 ∈ ℝ ) → { 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ∣ ( 𝐹 ‘ 𝑦 ) ≤ 𝑐 } ∈ ( 𝑆 ↾t 𝐷 ) ) |
48 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑏 ∈ ℝ ) → 𝑏 ∈ ℝ ) |
49 |
18 19 27 28 35 47 48
|
salpreimalegt |
⊢ ( ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑏 ∈ ℝ ) → { 𝑦 ∈ ∪ ( 𝑆 ↾t 𝐷 ) ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) |
50 |
13 49
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑏 ∈ ℝ ) → { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) |
51 |
50
|
ex |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ( 𝑏 ∈ ℝ → { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
52 |
9 51
|
ralrimi |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) |
53 |
5 6 52
|
3jca |
⊢ ( ( 𝜑 ∧ 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
54 |
53
|
ex |
⊢ ( 𝜑 → ( 𝐹 ∈ ( SMblFn ‘ 𝑆 ) → ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) ) ) |
55 |
|
nfv |
⊢ Ⅎ 𝑦 𝐷 ⊆ ∪ 𝑆 |
56 |
|
nfv |
⊢ Ⅎ 𝑦 𝐹 : 𝐷 ⟶ ℝ |
57 |
|
nfcv |
⊢ Ⅎ 𝑦 ℝ |
58 |
|
nfrab1 |
⊢ Ⅎ 𝑦 { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } |
59 |
|
nfcv |
⊢ Ⅎ 𝑦 ( 𝑆 ↾t 𝐷 ) |
60 |
58 59
|
nfel |
⊢ Ⅎ 𝑦 { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) |
61 |
57 60
|
nfralw |
⊢ Ⅎ 𝑦 ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) |
62 |
55 56 61
|
nf3an |
⊢ Ⅎ 𝑦 ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) |
63 |
14 62
|
nfan |
⊢ Ⅎ 𝑦 ( 𝜑 ∧ ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
64 |
|
nfv |
⊢ Ⅎ 𝑏 𝐷 ⊆ ∪ 𝑆 |
65 |
|
nfv |
⊢ Ⅎ 𝑏 𝐹 : 𝐷 ⟶ ℝ |
66 |
|
nfra1 |
⊢ Ⅎ 𝑏 ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) |
67 |
64 65 66
|
nf3an |
⊢ Ⅎ 𝑏 ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) |
68 |
7 67
|
nfan |
⊢ Ⅎ 𝑏 ( 𝜑 ∧ ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
69 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) ) → 𝑆 ∈ SAlg ) |
70 |
|
simpr1 |
⊢ ( ( 𝜑 ∧ ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) ) → 𝐷 ⊆ ∪ 𝑆 ) |
71 |
|
simpr2 |
⊢ ( ( 𝜑 ∧ ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) ) → 𝐹 : 𝐷 ⟶ ℝ ) |
72 |
|
simpr3 |
⊢ ( ( 𝜑 ∧ ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) ) → ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) |
73 |
63 68 69 2 70 71 72
|
issmfgtlem |
⊢ ( ( 𝜑 ∧ ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) ) → 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) |
74 |
73
|
ex |
⊢ ( 𝜑 → ( ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) → 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ) |
75 |
54 74
|
impbid |
⊢ ( 𝜑 → ( 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ↔ ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) ) ) |
76 |
|
breq1 |
⊢ ( 𝑏 = 𝑎 → ( 𝑏 < ( 𝐹 ‘ 𝑦 ) ↔ 𝑎 < ( 𝐹 ‘ 𝑦 ) ) ) |
77 |
76
|
rabbidv |
⊢ ( 𝑏 = 𝑎 → { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } = { 𝑦 ∈ 𝐷 ∣ 𝑎 < ( 𝐹 ‘ 𝑦 ) } ) |
78 |
|
fveq2 |
⊢ ( 𝑦 = 𝑥 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑥 ) ) |
79 |
78
|
breq2d |
⊢ ( 𝑦 = 𝑥 → ( 𝑎 < ( 𝐹 ‘ 𝑦 ) ↔ 𝑎 < ( 𝐹 ‘ 𝑥 ) ) ) |
80 |
79
|
cbvrabv |
⊢ { 𝑦 ∈ 𝐷 ∣ 𝑎 < ( 𝐹 ‘ 𝑦 ) } = { 𝑥 ∈ 𝐷 ∣ 𝑎 < ( 𝐹 ‘ 𝑥 ) } |
81 |
80
|
a1i |
⊢ ( 𝑏 = 𝑎 → { 𝑦 ∈ 𝐷 ∣ 𝑎 < ( 𝐹 ‘ 𝑦 ) } = { 𝑥 ∈ 𝐷 ∣ 𝑎 < ( 𝐹 ‘ 𝑥 ) } ) |
82 |
77 81
|
eqtrd |
⊢ ( 𝑏 = 𝑎 → { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } = { 𝑥 ∈ 𝐷 ∣ 𝑎 < ( 𝐹 ‘ 𝑥 ) } ) |
83 |
82
|
eleq1d |
⊢ ( 𝑏 = 𝑎 → ( { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ↔ { 𝑥 ∈ 𝐷 ∣ 𝑎 < ( 𝐹 ‘ 𝑥 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
84 |
83
|
cbvralvw |
⊢ ( ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ↔ ∀ 𝑎 ∈ ℝ { 𝑥 ∈ 𝐷 ∣ 𝑎 < ( 𝐹 ‘ 𝑥 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) |
85 |
84
|
3anbi3i |
⊢ ( ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) ↔ ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑎 ∈ ℝ { 𝑥 ∈ 𝐷 ∣ 𝑎 < ( 𝐹 ‘ 𝑥 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
86 |
85
|
a1i |
⊢ ( 𝜑 → ( ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦 ∈ 𝐷 ∣ 𝑏 < ( 𝐹 ‘ 𝑦 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) ↔ ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑎 ∈ ℝ { 𝑥 ∈ 𝐷 ∣ 𝑎 < ( 𝐹 ‘ 𝑥 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) ) ) |
87 |
75 86
|
bitrd |
⊢ ( 𝜑 → ( 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ↔ ( 𝐷 ⊆ ∪ 𝑆 ∧ 𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑎 ∈ ℝ { 𝑥 ∈ 𝐷 ∣ 𝑎 < ( 𝐹 ‘ 𝑥 ) } ∈ ( 𝑆 ↾t 𝐷 ) ) ) ) |