Step |
Hyp |
Ref |
Expression |
1 |
|
smfresal.s |
⊢ ( 𝜑 → 𝑆 ∈ SAlg ) |
2 |
|
smfresal.f |
⊢ ( 𝜑 → 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) |
3 |
|
smfresal.d |
⊢ 𝐷 = dom 𝐹 |
4 |
|
smfresal.t |
⊢ 𝑇 = { 𝑒 ∈ 𝒫 ℝ ∣ ( ◡ 𝐹 “ 𝑒 ) ∈ ( 𝑆 ↾t 𝐷 ) } |
5 |
|
reex |
⊢ ℝ ∈ V |
6 |
5
|
pwex |
⊢ 𝒫 ℝ ∈ V |
7 |
4 6
|
rabex2 |
⊢ 𝑇 ∈ V |
8 |
7
|
a1i |
⊢ ( 𝜑 → 𝑇 ∈ V ) |
9 |
|
0elpw |
⊢ ∅ ∈ 𝒫 ℝ |
10 |
9
|
a1i |
⊢ ( 𝜑 → ∅ ∈ 𝒫 ℝ ) |
11 |
|
ima0 |
⊢ ( ◡ 𝐹 “ ∅ ) = ∅ |
12 |
11
|
a1i |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ∅ ) = ∅ ) |
13 |
1
|
uniexd |
⊢ ( 𝜑 → ∪ 𝑆 ∈ V ) |
14 |
1 2 3
|
smfdmss |
⊢ ( 𝜑 → 𝐷 ⊆ ∪ 𝑆 ) |
15 |
13 14
|
ssexd |
⊢ ( 𝜑 → 𝐷 ∈ V ) |
16 |
|
eqid |
⊢ ( 𝑆 ↾t 𝐷 ) = ( 𝑆 ↾t 𝐷 ) |
17 |
1 15 16
|
subsalsal |
⊢ ( 𝜑 → ( 𝑆 ↾t 𝐷 ) ∈ SAlg ) |
18 |
17
|
0sald |
⊢ ( 𝜑 → ∅ ∈ ( 𝑆 ↾t 𝐷 ) ) |
19 |
12 18
|
eqeltrd |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ∅ ) ∈ ( 𝑆 ↾t 𝐷 ) ) |
20 |
10 19
|
jca |
⊢ ( 𝜑 → ( ∅ ∈ 𝒫 ℝ ∧ ( ◡ 𝐹 “ ∅ ) ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
21 |
|
imaeq2 |
⊢ ( 𝑒 = ∅ → ( ◡ 𝐹 “ 𝑒 ) = ( ◡ 𝐹 “ ∅ ) ) |
22 |
21
|
eleq1d |
⊢ ( 𝑒 = ∅ → ( ( ◡ 𝐹 “ 𝑒 ) ∈ ( 𝑆 ↾t 𝐷 ) ↔ ( ◡ 𝐹 “ ∅ ) ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
23 |
22 4
|
elrab2 |
⊢ ( ∅ ∈ 𝑇 ↔ ( ∅ ∈ 𝒫 ℝ ∧ ( ◡ 𝐹 “ ∅ ) ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
24 |
20 23
|
sylibr |
⊢ ( 𝜑 → ∅ ∈ 𝑇 ) |
25 |
|
eqid |
⊢ ∪ 𝑇 = ∪ 𝑇 |
26 |
|
nfv |
⊢ Ⅎ 𝑦 𝜑 |
27 |
|
nfcv |
⊢ Ⅎ 𝑒 𝑦 |
28 |
|
nfrab1 |
⊢ Ⅎ 𝑒 { 𝑒 ∈ 𝒫 ℝ ∣ ( ◡ 𝐹 “ 𝑒 ) ∈ ( 𝑆 ↾t 𝐷 ) } |
29 |
4 28
|
nfcxfr |
⊢ Ⅎ 𝑒 𝑇 |
30 |
27 29
|
eluni2f |
⊢ ( 𝑦 ∈ ∪ 𝑇 ↔ ∃ 𝑒 ∈ 𝑇 𝑦 ∈ 𝑒 ) |
31 |
30
|
biimpi |
⊢ ( 𝑦 ∈ ∪ 𝑇 → ∃ 𝑒 ∈ 𝑇 𝑦 ∈ 𝑒 ) |
32 |
31
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ∪ 𝑇 ) → ∃ 𝑒 ∈ 𝑇 𝑦 ∈ 𝑒 ) |
33 |
|
nfv |
⊢ Ⅎ 𝑒 𝜑 |
34 |
29
|
nfuni |
⊢ Ⅎ 𝑒 ∪ 𝑇 |
35 |
27 34
|
nfel |
⊢ Ⅎ 𝑒 𝑦 ∈ ∪ 𝑇 |
36 |
33 35
|
nfan |
⊢ Ⅎ 𝑒 ( 𝜑 ∧ 𝑦 ∈ ∪ 𝑇 ) |
37 |
27
|
nfel1 |
⊢ Ⅎ 𝑒 𝑦 ∈ ℝ |
38 |
4
|
eleq2i |
⊢ ( 𝑒 ∈ 𝑇 ↔ 𝑒 ∈ { 𝑒 ∈ 𝒫 ℝ ∣ ( ◡ 𝐹 “ 𝑒 ) ∈ ( 𝑆 ↾t 𝐷 ) } ) |
39 |
38
|
biimpi |
⊢ ( 𝑒 ∈ 𝑇 → 𝑒 ∈ { 𝑒 ∈ 𝒫 ℝ ∣ ( ◡ 𝐹 “ 𝑒 ) ∈ ( 𝑆 ↾t 𝐷 ) } ) |
40 |
|
rabidim1 |
⊢ ( 𝑒 ∈ { 𝑒 ∈ 𝒫 ℝ ∣ ( ◡ 𝐹 “ 𝑒 ) ∈ ( 𝑆 ↾t 𝐷 ) } → 𝑒 ∈ 𝒫 ℝ ) |
41 |
39 40
|
syl |
⊢ ( 𝑒 ∈ 𝑇 → 𝑒 ∈ 𝒫 ℝ ) |
42 |
|
elpwi |
⊢ ( 𝑒 ∈ 𝒫 ℝ → 𝑒 ⊆ ℝ ) |
43 |
41 42
|
syl |
⊢ ( 𝑒 ∈ 𝑇 → 𝑒 ⊆ ℝ ) |
44 |
43
|
adantr |
⊢ ( ( 𝑒 ∈ 𝑇 ∧ 𝑦 ∈ 𝑒 ) → 𝑒 ⊆ ℝ ) |
45 |
|
simpr |
⊢ ( ( 𝑒 ∈ 𝑇 ∧ 𝑦 ∈ 𝑒 ) → 𝑦 ∈ 𝑒 ) |
46 |
44 45
|
sseldd |
⊢ ( ( 𝑒 ∈ 𝑇 ∧ 𝑦 ∈ 𝑒 ) → 𝑦 ∈ ℝ ) |
47 |
46
|
ex |
⊢ ( 𝑒 ∈ 𝑇 → ( 𝑦 ∈ 𝑒 → 𝑦 ∈ ℝ ) ) |
48 |
47
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ∪ 𝑇 ) → ( 𝑒 ∈ 𝑇 → ( 𝑦 ∈ 𝑒 → 𝑦 ∈ ℝ ) ) ) |
49 |
36 37 48
|
rexlimd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ∪ 𝑇 ) → ( ∃ 𝑒 ∈ 𝑇 𝑦 ∈ 𝑒 → 𝑦 ∈ ℝ ) ) |
50 |
32 49
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ∪ 𝑇 ) → 𝑦 ∈ ℝ ) |
51 |
50
|
ex |
⊢ ( 𝜑 → ( 𝑦 ∈ ∪ 𝑇 → 𝑦 ∈ ℝ ) ) |
52 |
|
ovexd |
⊢ ( 𝜑 → ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ∈ V ) |
53 |
|
ioossre |
⊢ ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ⊆ ℝ |
54 |
53
|
a1i |
⊢ ( 𝜑 → ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ⊆ ℝ ) |
55 |
52 54
|
elpwd |
⊢ ( 𝜑 → ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ∈ 𝒫 ℝ ) |
56 |
55
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ∈ 𝒫 ℝ ) |
57 |
1 2 3
|
smff |
⊢ ( 𝜑 → 𝐹 : 𝐷 ⟶ ℝ ) |
58 |
57
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn 𝐷 ) |
59 |
|
fncnvima2 |
⊢ ( 𝐹 Fn 𝐷 → ( ◡ 𝐹 “ ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ) = { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) } ) |
60 |
58 59
|
syl |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ) = { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) } ) |
61 |
60
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ◡ 𝐹 “ ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ) = { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) } ) |
62 |
|
nfv |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑦 ∈ ℝ ) |
63 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑆 ∈ SAlg ) |
64 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐷 ∈ V ) |
65 |
57
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → 𝐹 : 𝐷 ⟶ ℝ ) |
66 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → 𝑥 ∈ 𝐷 ) |
67 |
65 66
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → ( 𝐹 ‘ 𝑥 ) ∈ ℝ ) |
68 |
67
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑥 ∈ 𝐷 ) → ( 𝐹 ‘ 𝑥 ) ∈ ℝ ) |
69 |
57
|
feqmptd |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐷 ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
70 |
69
|
eqcomd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐷 ↦ ( 𝐹 ‘ 𝑥 ) ) = 𝐹 ) |
71 |
70 2
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐷 ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ ( SMblFn ‘ 𝑆 ) ) |
72 |
71
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑥 ∈ 𝐷 ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ ( SMblFn ‘ 𝑆 ) ) |
73 |
|
peano2rem |
⊢ ( 𝑦 ∈ ℝ → ( 𝑦 − 1 ) ∈ ℝ ) |
74 |
73
|
rexrd |
⊢ ( 𝑦 ∈ ℝ → ( 𝑦 − 1 ) ∈ ℝ* ) |
75 |
74
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑦 − 1 ) ∈ ℝ* ) |
76 |
|
peano2re |
⊢ ( 𝑦 ∈ ℝ → ( 𝑦 + 1 ) ∈ ℝ ) |
77 |
76
|
rexrd |
⊢ ( 𝑦 ∈ ℝ → ( 𝑦 + 1 ) ∈ ℝ* ) |
78 |
77
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑦 + 1 ) ∈ ℝ* ) |
79 |
62 63 64 68 72 75 78
|
smfpimioompt |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → { 𝑥 ∈ 𝐷 ∣ ( 𝐹 ‘ 𝑥 ) ∈ ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) } ∈ ( 𝑆 ↾t 𝐷 ) ) |
80 |
61 79
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ◡ 𝐹 “ ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) |
81 |
56 80
|
jca |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ∈ 𝒫 ℝ ∧ ( ◡ 𝐹 “ ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
82 |
|
imaeq2 |
⊢ ( 𝑒 = ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) → ( ◡ 𝐹 “ 𝑒 ) = ( ◡ 𝐹 “ ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ) ) |
83 |
82
|
eleq1d |
⊢ ( 𝑒 = ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) → ( ( ◡ 𝐹 “ 𝑒 ) ∈ ( 𝑆 ↾t 𝐷 ) ↔ ( ◡ 𝐹 “ ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
84 |
83 4
|
elrab2 |
⊢ ( ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ∈ 𝑇 ↔ ( ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ∈ 𝒫 ℝ ∧ ( ◡ 𝐹 “ ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
85 |
81 84
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ∈ 𝑇 ) |
86 |
|
id |
⊢ ( 𝑦 ∈ ℝ → 𝑦 ∈ ℝ ) |
87 |
|
ltm1 |
⊢ ( 𝑦 ∈ ℝ → ( 𝑦 − 1 ) < 𝑦 ) |
88 |
|
ltp1 |
⊢ ( 𝑦 ∈ ℝ → 𝑦 < ( 𝑦 + 1 ) ) |
89 |
74 77 86 87 88
|
eliood |
⊢ ( 𝑦 ∈ ℝ → 𝑦 ∈ ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ) |
90 |
89
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑦 ∈ ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ) |
91 |
|
nfv |
⊢ Ⅎ 𝑒 𝑦 ∈ ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) |
92 |
|
nfcv |
⊢ Ⅎ 𝑒 ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) |
93 |
|
eleq2 |
⊢ ( 𝑒 = ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) → ( 𝑦 ∈ 𝑒 ↔ 𝑦 ∈ ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ) ) |
94 |
91 92 29 93
|
rspcef |
⊢ ( ( ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ∈ 𝑇 ∧ 𝑦 ∈ ( ( 𝑦 − 1 ) (,) ( 𝑦 + 1 ) ) ) → ∃ 𝑒 ∈ 𝑇 𝑦 ∈ 𝑒 ) |
95 |
85 90 94
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ∃ 𝑒 ∈ 𝑇 𝑦 ∈ 𝑒 ) |
96 |
95 30
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑦 ∈ ∪ 𝑇 ) |
97 |
96
|
ex |
⊢ ( 𝜑 → ( 𝑦 ∈ ℝ → 𝑦 ∈ ∪ 𝑇 ) ) |
98 |
51 97
|
impbid |
⊢ ( 𝜑 → ( 𝑦 ∈ ∪ 𝑇 ↔ 𝑦 ∈ ℝ ) ) |
99 |
26 98
|
alrimi |
⊢ ( 𝜑 → ∀ 𝑦 ( 𝑦 ∈ ∪ 𝑇 ↔ 𝑦 ∈ ℝ ) ) |
100 |
|
dfcleq |
⊢ ( ∪ 𝑇 = ℝ ↔ ∀ 𝑦 ( 𝑦 ∈ ∪ 𝑇 ↔ 𝑦 ∈ ℝ ) ) |
101 |
99 100
|
sylibr |
⊢ ( 𝜑 → ∪ 𝑇 = ℝ ) |
102 |
101
|
difeq1d |
⊢ ( 𝜑 → ( ∪ 𝑇 ∖ 𝑥 ) = ( ℝ ∖ 𝑥 ) ) |
103 |
102
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑇 ) → ( ∪ 𝑇 ∖ 𝑥 ) = ( ℝ ∖ 𝑥 ) ) |
104 |
|
difss |
⊢ ( ℝ ∖ 𝑥 ) ⊆ ℝ |
105 |
5 104
|
ssexi |
⊢ ( ℝ ∖ 𝑥 ) ∈ V |
106 |
|
elpwg |
⊢ ( ( ℝ ∖ 𝑥 ) ∈ V → ( ( ℝ ∖ 𝑥 ) ∈ 𝒫 ℝ ↔ ( ℝ ∖ 𝑥 ) ⊆ ℝ ) ) |
107 |
105 106
|
ax-mp |
⊢ ( ( ℝ ∖ 𝑥 ) ∈ 𝒫 ℝ ↔ ( ℝ ∖ 𝑥 ) ⊆ ℝ ) |
108 |
104 107
|
mpbir |
⊢ ( ℝ ∖ 𝑥 ) ∈ 𝒫 ℝ |
109 |
108
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑇 ) → ( ℝ ∖ 𝑥 ) ∈ 𝒫 ℝ ) |
110 |
57
|
ffund |
⊢ ( 𝜑 → Fun 𝐹 ) |
111 |
|
difpreima |
⊢ ( Fun 𝐹 → ( ◡ 𝐹 “ ( ℝ ∖ 𝑥 ) ) = ( ( ◡ 𝐹 “ ℝ ) ∖ ( ◡ 𝐹 “ 𝑥 ) ) ) |
112 |
110 111
|
syl |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ( ℝ ∖ 𝑥 ) ) = ( ( ◡ 𝐹 “ ℝ ) ∖ ( ◡ 𝐹 “ 𝑥 ) ) ) |
113 |
|
fimacnv |
⊢ ( 𝐹 : 𝐷 ⟶ ℝ → ( ◡ 𝐹 “ ℝ ) = 𝐷 ) |
114 |
57 113
|
syl |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ℝ ) = 𝐷 ) |
115 |
1 14
|
restuni4 |
⊢ ( 𝜑 → ∪ ( 𝑆 ↾t 𝐷 ) = 𝐷 ) |
116 |
114 115
|
eqtr4d |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ℝ ) = ∪ ( 𝑆 ↾t 𝐷 ) ) |
117 |
116
|
difeq1d |
⊢ ( 𝜑 → ( ( ◡ 𝐹 “ ℝ ) ∖ ( ◡ 𝐹 “ 𝑥 ) ) = ( ∪ ( 𝑆 ↾t 𝐷 ) ∖ ( ◡ 𝐹 “ 𝑥 ) ) ) |
118 |
112 117
|
eqtrd |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ( ℝ ∖ 𝑥 ) ) = ( ∪ ( 𝑆 ↾t 𝐷 ) ∖ ( ◡ 𝐹 “ 𝑥 ) ) ) |
119 |
118
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑇 ) → ( ◡ 𝐹 “ ( ℝ ∖ 𝑥 ) ) = ( ∪ ( 𝑆 ↾t 𝐷 ) ∖ ( ◡ 𝐹 “ 𝑥 ) ) ) |
120 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑇 ) → ( 𝑆 ↾t 𝐷 ) ∈ SAlg ) |
121 |
|
imaeq2 |
⊢ ( 𝑒 = 𝑥 → ( ◡ 𝐹 “ 𝑒 ) = ( ◡ 𝐹 “ 𝑥 ) ) |
122 |
121
|
eleq1d |
⊢ ( 𝑒 = 𝑥 → ( ( ◡ 𝐹 “ 𝑒 ) ∈ ( 𝑆 ↾t 𝐷 ) ↔ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
123 |
122 4
|
elrab2 |
⊢ ( 𝑥 ∈ 𝑇 ↔ ( 𝑥 ∈ 𝒫 ℝ ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
124 |
123
|
biimpi |
⊢ ( 𝑥 ∈ 𝑇 → ( 𝑥 ∈ 𝒫 ℝ ∧ ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
125 |
124
|
simprd |
⊢ ( 𝑥 ∈ 𝑇 → ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝑆 ↾t 𝐷 ) ) |
126 |
125
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑇 ) → ( ◡ 𝐹 “ 𝑥 ) ∈ ( 𝑆 ↾t 𝐷 ) ) |
127 |
120 126
|
saldifcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑇 ) → ( ∪ ( 𝑆 ↾t 𝐷 ) ∖ ( ◡ 𝐹 “ 𝑥 ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) |
128 |
119 127
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑇 ) → ( ◡ 𝐹 “ ( ℝ ∖ 𝑥 ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) |
129 |
109 128
|
jca |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑇 ) → ( ( ℝ ∖ 𝑥 ) ∈ 𝒫 ℝ ∧ ( ◡ 𝐹 “ ( ℝ ∖ 𝑥 ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
130 |
|
imaeq2 |
⊢ ( 𝑒 = ( ℝ ∖ 𝑥 ) → ( ◡ 𝐹 “ 𝑒 ) = ( ◡ 𝐹 “ ( ℝ ∖ 𝑥 ) ) ) |
131 |
130
|
eleq1d |
⊢ ( 𝑒 = ( ℝ ∖ 𝑥 ) → ( ( ◡ 𝐹 “ 𝑒 ) ∈ ( 𝑆 ↾t 𝐷 ) ↔ ( ◡ 𝐹 “ ( ℝ ∖ 𝑥 ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
132 |
131 4
|
elrab2 |
⊢ ( ( ℝ ∖ 𝑥 ) ∈ 𝑇 ↔ ( ( ℝ ∖ 𝑥 ) ∈ 𝒫 ℝ ∧ ( ◡ 𝐹 “ ( ℝ ∖ 𝑥 ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
133 |
129 132
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑇 ) → ( ℝ ∖ 𝑥 ) ∈ 𝑇 ) |
134 |
103 133
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑇 ) → ( ∪ 𝑇 ∖ 𝑥 ) ∈ 𝑇 ) |
135 |
|
nnex |
⊢ ℕ ∈ V |
136 |
|
fvex |
⊢ ( 𝑔 ‘ 𝑛 ) ∈ V |
137 |
135 136
|
iunex |
⊢ ∪ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) ∈ V |
138 |
137
|
a1i |
⊢ ( 𝑔 : ℕ ⟶ 𝑇 → ∪ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) ∈ V ) |
139 |
|
ffvelrn |
⊢ ( ( 𝑔 : ℕ ⟶ 𝑇 ∧ 𝑛 ∈ ℕ ) → ( 𝑔 ‘ 𝑛 ) ∈ 𝑇 ) |
140 |
4
|
eleq2i |
⊢ ( ( 𝑔 ‘ 𝑛 ) ∈ 𝑇 ↔ ( 𝑔 ‘ 𝑛 ) ∈ { 𝑒 ∈ 𝒫 ℝ ∣ ( ◡ 𝐹 “ 𝑒 ) ∈ ( 𝑆 ↾t 𝐷 ) } ) |
141 |
140
|
biimpi |
⊢ ( ( 𝑔 ‘ 𝑛 ) ∈ 𝑇 → ( 𝑔 ‘ 𝑛 ) ∈ { 𝑒 ∈ 𝒫 ℝ ∣ ( ◡ 𝐹 “ 𝑒 ) ∈ ( 𝑆 ↾t 𝐷 ) } ) |
142 |
|
elrabi |
⊢ ( ( 𝑔 ‘ 𝑛 ) ∈ { 𝑒 ∈ 𝒫 ℝ ∣ ( ◡ 𝐹 “ 𝑒 ) ∈ ( 𝑆 ↾t 𝐷 ) } → ( 𝑔 ‘ 𝑛 ) ∈ 𝒫 ℝ ) |
143 |
141 142
|
syl |
⊢ ( ( 𝑔 ‘ 𝑛 ) ∈ 𝑇 → ( 𝑔 ‘ 𝑛 ) ∈ 𝒫 ℝ ) |
144 |
|
elpwi |
⊢ ( ( 𝑔 ‘ 𝑛 ) ∈ 𝒫 ℝ → ( 𝑔 ‘ 𝑛 ) ⊆ ℝ ) |
145 |
143 144
|
syl |
⊢ ( ( 𝑔 ‘ 𝑛 ) ∈ 𝑇 → ( 𝑔 ‘ 𝑛 ) ⊆ ℝ ) |
146 |
139 145
|
syl |
⊢ ( ( 𝑔 : ℕ ⟶ 𝑇 ∧ 𝑛 ∈ ℕ ) → ( 𝑔 ‘ 𝑛 ) ⊆ ℝ ) |
147 |
146
|
iunssd |
⊢ ( 𝑔 : ℕ ⟶ 𝑇 → ∪ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) ⊆ ℝ ) |
148 |
138 147
|
elpwd |
⊢ ( 𝑔 : ℕ ⟶ 𝑇 → ∪ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) ∈ 𝒫 ℝ ) |
149 |
148
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ 𝑇 ) → ∪ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) ∈ 𝒫 ℝ ) |
150 |
|
imaiun |
⊢ ( ◡ 𝐹 “ ∪ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) ) = ∪ 𝑛 ∈ ℕ ( ◡ 𝐹 “ ( 𝑔 ‘ 𝑛 ) ) |
151 |
150
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ 𝑇 ) → ( ◡ 𝐹 “ ∪ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) ) = ∪ 𝑛 ∈ ℕ ( ◡ 𝐹 “ ( 𝑔 ‘ 𝑛 ) ) ) |
152 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ 𝑇 ) → ( 𝑆 ↾t 𝐷 ) ∈ SAlg ) |
153 |
|
nnct |
⊢ ℕ ≼ ω |
154 |
153
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ 𝑇 ) → ℕ ≼ ω ) |
155 |
|
imaeq2 |
⊢ ( 𝑒 = ( 𝑔 ‘ 𝑛 ) → ( ◡ 𝐹 “ 𝑒 ) = ( ◡ 𝐹 “ ( 𝑔 ‘ 𝑛 ) ) ) |
156 |
155
|
eleq1d |
⊢ ( 𝑒 = ( 𝑔 ‘ 𝑛 ) → ( ( ◡ 𝐹 “ 𝑒 ) ∈ ( 𝑆 ↾t 𝐷 ) ↔ ( ◡ 𝐹 “ ( 𝑔 ‘ 𝑛 ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
157 |
156 4
|
elrab2 |
⊢ ( ( 𝑔 ‘ 𝑛 ) ∈ 𝑇 ↔ ( ( 𝑔 ‘ 𝑛 ) ∈ 𝒫 ℝ ∧ ( ◡ 𝐹 “ ( 𝑔 ‘ 𝑛 ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
158 |
157
|
biimpi |
⊢ ( ( 𝑔 ‘ 𝑛 ) ∈ 𝑇 → ( ( 𝑔 ‘ 𝑛 ) ∈ 𝒫 ℝ ∧ ( ◡ 𝐹 “ ( 𝑔 ‘ 𝑛 ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
159 |
158
|
simprd |
⊢ ( ( 𝑔 ‘ 𝑛 ) ∈ 𝑇 → ( ◡ 𝐹 “ ( 𝑔 ‘ 𝑛 ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) |
160 |
139 159
|
syl |
⊢ ( ( 𝑔 : ℕ ⟶ 𝑇 ∧ 𝑛 ∈ ℕ ) → ( ◡ 𝐹 “ ( 𝑔 ‘ 𝑛 ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) |
161 |
160
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ 𝑇 ) ∧ 𝑛 ∈ ℕ ) → ( ◡ 𝐹 “ ( 𝑔 ‘ 𝑛 ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) |
162 |
152 154 161
|
saliuncl |
⊢ ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ 𝑇 ) → ∪ 𝑛 ∈ ℕ ( ◡ 𝐹 “ ( 𝑔 ‘ 𝑛 ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) |
163 |
151 162
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ 𝑇 ) → ( ◡ 𝐹 “ ∪ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) |
164 |
149 163
|
jca |
⊢ ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ 𝑇 ) → ( ∪ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) ∈ 𝒫 ℝ ∧ ( ◡ 𝐹 “ ∪ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
165 |
|
imaeq2 |
⊢ ( 𝑒 = ∪ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) → ( ◡ 𝐹 “ 𝑒 ) = ( ◡ 𝐹 “ ∪ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) ) ) |
166 |
165
|
eleq1d |
⊢ ( 𝑒 = ∪ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) → ( ( ◡ 𝐹 “ 𝑒 ) ∈ ( 𝑆 ↾t 𝐷 ) ↔ ( ◡ 𝐹 “ ∪ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
167 |
166 4
|
elrab2 |
⊢ ( ∪ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) ∈ 𝑇 ↔ ( ∪ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) ∈ 𝒫 ℝ ∧ ( ◡ 𝐹 “ ∪ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) ) ∈ ( 𝑆 ↾t 𝐷 ) ) ) |
168 |
164 167
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑔 : ℕ ⟶ 𝑇 ) → ∪ 𝑛 ∈ ℕ ( 𝑔 ‘ 𝑛 ) ∈ 𝑇 ) |
169 |
8 24 25 134 168
|
issalnnd |
⊢ ( 𝜑 → 𝑇 ∈ SAlg ) |