Step |
Hyp |
Ref |
Expression |
1 |
|
ismrcd.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑉 ) |
2 |
|
ismrcd.f |
⊢ ( 𝜑 → 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ) |
3 |
|
ismrcd.e |
⊢ ( ( 𝜑 ∧ 𝑥 ⊆ 𝐵 ) → 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ) |
4 |
|
ismrcd.m |
⊢ ( ( 𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) |
5 |
|
ismrcd.i |
⊢ ( ( 𝜑 ∧ 𝑥 ⊆ 𝐵 ) → ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
6 |
|
inss1 |
⊢ ( 𝐹 ∩ I ) ⊆ 𝐹 |
7 |
|
dmss |
⊢ ( ( 𝐹 ∩ I ) ⊆ 𝐹 → dom ( 𝐹 ∩ I ) ⊆ dom 𝐹 ) |
8 |
6 7
|
ax-mp |
⊢ dom ( 𝐹 ∩ I ) ⊆ dom 𝐹 |
9 |
8 2
|
fssdm |
⊢ ( 𝜑 → dom ( 𝐹 ∩ I ) ⊆ 𝒫 𝐵 ) |
10 |
|
ssid |
⊢ 𝐵 ⊆ 𝐵 |
11 |
|
elpwg |
⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∈ 𝒫 𝐵 ↔ 𝐵 ⊆ 𝐵 ) ) |
12 |
1 11
|
syl |
⊢ ( 𝜑 → ( 𝐵 ∈ 𝒫 𝐵 ↔ 𝐵 ⊆ 𝐵 ) ) |
13 |
10 12
|
mpbiri |
⊢ ( 𝜑 → 𝐵 ∈ 𝒫 𝐵 ) |
14 |
2 13
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ∈ 𝒫 𝐵 ) |
15 |
14
|
elpwid |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ⊆ 𝐵 ) |
16 |
|
velpw |
⊢ ( 𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵 ) |
17 |
16 3
|
sylan2b |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝒫 𝐵 ) → 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ) |
18 |
17
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝒫 𝐵 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ) |
19 |
|
id |
⊢ ( 𝑥 = 𝐵 → 𝑥 = 𝐵 ) |
20 |
|
fveq2 |
⊢ ( 𝑥 = 𝐵 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝐵 ) ) |
21 |
19 20
|
sseq12d |
⊢ ( 𝑥 = 𝐵 → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ↔ 𝐵 ⊆ ( 𝐹 ‘ 𝐵 ) ) ) |
22 |
21
|
rspcva |
⊢ ( ( 𝐵 ∈ 𝒫 𝐵 ∧ ∀ 𝑥 ∈ 𝒫 𝐵 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ) → 𝐵 ⊆ ( 𝐹 ‘ 𝐵 ) ) |
23 |
13 18 22
|
syl2anc |
⊢ ( 𝜑 → 𝐵 ⊆ ( 𝐹 ‘ 𝐵 ) ) |
24 |
15 23
|
eqssd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) = 𝐵 ) |
25 |
2
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn 𝒫 𝐵 ) |
26 |
|
fnelfp |
⊢ ( ( 𝐹 Fn 𝒫 𝐵 ∧ 𝐵 ∈ 𝒫 𝐵 ) → ( 𝐵 ∈ dom ( 𝐹 ∩ I ) ↔ ( 𝐹 ‘ 𝐵 ) = 𝐵 ) ) |
27 |
25 13 26
|
syl2anc |
⊢ ( 𝜑 → ( 𝐵 ∈ dom ( 𝐹 ∩ I ) ↔ ( 𝐹 ‘ 𝐵 ) = 𝐵 ) ) |
28 |
24 27
|
mpbird |
⊢ ( 𝜑 → 𝐵 ∈ dom ( 𝐹 ∩ I ) ) |
29 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → 𝑧 ⊆ dom ( 𝐹 ∩ I ) ) |
30 |
9
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → dom ( 𝐹 ∩ I ) ⊆ 𝒫 𝐵 ) |
31 |
29 30
|
sstrd |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → 𝑧 ⊆ 𝒫 𝐵 ) |
32 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → 𝑧 ≠ ∅ ) |
33 |
|
intssuni2 |
⊢ ( ( 𝑧 ⊆ 𝒫 𝐵 ∧ 𝑧 ≠ ∅ ) → ∩ 𝑧 ⊆ ∪ 𝒫 𝐵 ) |
34 |
31 32 33
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → ∩ 𝑧 ⊆ ∪ 𝒫 𝐵 ) |
35 |
|
unipw |
⊢ ∪ 𝒫 𝐵 = 𝐵 |
36 |
34 35
|
sseqtrdi |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → ∩ 𝑧 ⊆ 𝐵 ) |
37 |
|
intex |
⊢ ( 𝑧 ≠ ∅ ↔ ∩ 𝑧 ∈ V ) |
38 |
|
elpwg |
⊢ ( ∩ 𝑧 ∈ V → ( ∩ 𝑧 ∈ 𝒫 𝐵 ↔ ∩ 𝑧 ⊆ 𝐵 ) ) |
39 |
37 38
|
sylbi |
⊢ ( 𝑧 ≠ ∅ → ( ∩ 𝑧 ∈ 𝒫 𝐵 ↔ ∩ 𝑧 ⊆ 𝐵 ) ) |
40 |
39
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → ( ∩ 𝑧 ∈ 𝒫 𝐵 ↔ ∩ 𝑧 ⊆ 𝐵 ) ) |
41 |
36 40
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → ∩ 𝑧 ∈ 𝒫 𝐵 ) |
42 |
41
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥 ∈ 𝑧 ) → ∩ 𝑧 ∈ 𝒫 𝐵 ) |
43 |
4
|
3expib |
⊢ ( 𝜑 → ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) ) |
44 |
43
|
alrimiv |
⊢ ( 𝜑 → ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) ) |
45 |
44
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) ) |
46 |
45
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥 ∈ 𝑧 ) → ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) ) |
47 |
31
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥 ∈ 𝑧 ) → 𝑥 ∈ 𝒫 𝐵 ) |
48 |
47
|
elpwid |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥 ∈ 𝑧 ) → 𝑥 ⊆ 𝐵 ) |
49 |
|
intss1 |
⊢ ( 𝑥 ∈ 𝑧 → ∩ 𝑧 ⊆ 𝑥 ) |
50 |
49
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥 ∈ 𝑧 ) → ∩ 𝑧 ⊆ 𝑥 ) |
51 |
48 50
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥 ∈ 𝑧 ) → ( 𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥 ) ) |
52 |
|
sseq1 |
⊢ ( 𝑦 = ∩ 𝑧 → ( 𝑦 ⊆ 𝑥 ↔ ∩ 𝑧 ⊆ 𝑥 ) ) |
53 |
52
|
anbi2d |
⊢ ( 𝑦 = ∩ 𝑧 → ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) ↔ ( 𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥 ) ) ) |
54 |
|
fveq2 |
⊢ ( 𝑦 = ∩ 𝑧 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ ∩ 𝑧 ) ) |
55 |
54
|
sseq1d |
⊢ ( 𝑦 = ∩ 𝑧 → ( ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ ∩ 𝑧 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) ) |
56 |
53 55
|
imbi12d |
⊢ ( 𝑦 = ∩ 𝑧 → ( ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) ↔ ( ( 𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥 ) → ( 𝐹 ‘ ∩ 𝑧 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) ) ) |
57 |
56
|
spcgv |
⊢ ( ∩ 𝑧 ∈ 𝒫 𝐵 → ( ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) → ( ( 𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥 ) → ( 𝐹 ‘ ∩ 𝑧 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) ) ) |
58 |
42 46 51 57
|
syl3c |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥 ∈ 𝑧 ) → ( 𝐹 ‘ ∩ 𝑧 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) |
59 |
29
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥 ∈ 𝑧 ) → 𝑥 ∈ dom ( 𝐹 ∩ I ) ) |
60 |
25
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → 𝐹 Fn 𝒫 𝐵 ) |
61 |
60
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥 ∈ 𝑧 ) → 𝐹 Fn 𝒫 𝐵 ) |
62 |
|
fnelfp |
⊢ ( ( 𝐹 Fn 𝒫 𝐵 ∧ 𝑥 ∈ 𝒫 𝐵 ) → ( 𝑥 ∈ dom ( 𝐹 ∩ I ) ↔ ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) |
63 |
61 47 62
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥 ∈ 𝑧 ) → ( 𝑥 ∈ dom ( 𝐹 ∩ I ) ↔ ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) |
64 |
59 63
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥 ∈ 𝑧 ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) |
65 |
58 64
|
sseqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥 ∈ 𝑧 ) → ( 𝐹 ‘ ∩ 𝑧 ) ⊆ 𝑥 ) |
66 |
65
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → ∀ 𝑥 ∈ 𝑧 ( 𝐹 ‘ ∩ 𝑧 ) ⊆ 𝑥 ) |
67 |
|
ssint |
⊢ ( ( 𝐹 ‘ ∩ 𝑧 ) ⊆ ∩ 𝑧 ↔ ∀ 𝑥 ∈ 𝑧 ( 𝐹 ‘ ∩ 𝑧 ) ⊆ 𝑥 ) |
68 |
66 67
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → ( 𝐹 ‘ ∩ 𝑧 ) ⊆ ∩ 𝑧 ) |
69 |
18
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → ∀ 𝑥 ∈ 𝒫 𝐵 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ) |
70 |
|
id |
⊢ ( 𝑥 = ∩ 𝑧 → 𝑥 = ∩ 𝑧 ) |
71 |
|
fveq2 |
⊢ ( 𝑥 = ∩ 𝑧 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ ∩ 𝑧 ) ) |
72 |
70 71
|
sseq12d |
⊢ ( 𝑥 = ∩ 𝑧 → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ↔ ∩ 𝑧 ⊆ ( 𝐹 ‘ ∩ 𝑧 ) ) ) |
73 |
72
|
rspcva |
⊢ ( ( ∩ 𝑧 ∈ 𝒫 𝐵 ∧ ∀ 𝑥 ∈ 𝒫 𝐵 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ) → ∩ 𝑧 ⊆ ( 𝐹 ‘ ∩ 𝑧 ) ) |
74 |
41 69 73
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → ∩ 𝑧 ⊆ ( 𝐹 ‘ ∩ 𝑧 ) ) |
75 |
68 74
|
eqssd |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → ( 𝐹 ‘ ∩ 𝑧 ) = ∩ 𝑧 ) |
76 |
|
fnelfp |
⊢ ( ( 𝐹 Fn 𝒫 𝐵 ∧ ∩ 𝑧 ∈ 𝒫 𝐵 ) → ( ∩ 𝑧 ∈ dom ( 𝐹 ∩ I ) ↔ ( 𝐹 ‘ ∩ 𝑧 ) = ∩ 𝑧 ) ) |
77 |
60 41 76
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → ( ∩ 𝑧 ∈ dom ( 𝐹 ∩ I ) ↔ ( 𝐹 ‘ ∩ 𝑧 ) = ∩ 𝑧 ) ) |
78 |
75 77
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → ∩ 𝑧 ∈ dom ( 𝐹 ∩ I ) ) |
79 |
9 28 78
|
ismred |
⊢ ( 𝜑 → dom ( 𝐹 ∩ I ) ∈ ( Moore ‘ 𝐵 ) ) |