Step |
Hyp |
Ref |
Expression |
1 |
|
madebdayim |
⊢ ( 𝑋 ∈ ( M ‘ 𝐴 ) → ( bday ‘ 𝑋 ) ⊆ 𝐴 ) |
2 |
|
sseq2 |
⊢ ( 𝑎 = 𝑏 → ( ( bday ‘ 𝑥 ) ⊆ 𝑎 ↔ ( bday ‘ 𝑥 ) ⊆ 𝑏 ) ) |
3 |
|
fveq2 |
⊢ ( 𝑎 = 𝑏 → ( M ‘ 𝑎 ) = ( M ‘ 𝑏 ) ) |
4 |
3
|
eleq2d |
⊢ ( 𝑎 = 𝑏 → ( 𝑥 ∈ ( M ‘ 𝑎 ) ↔ 𝑥 ∈ ( M ‘ 𝑏 ) ) ) |
5 |
2 4
|
imbi12d |
⊢ ( 𝑎 = 𝑏 → ( ( ( bday ‘ 𝑥 ) ⊆ 𝑎 → 𝑥 ∈ ( M ‘ 𝑎 ) ) ↔ ( ( bday ‘ 𝑥 ) ⊆ 𝑏 → 𝑥 ∈ ( M ‘ 𝑏 ) ) ) ) |
6 |
5
|
ralbidv |
⊢ ( 𝑎 = 𝑏 → ( ∀ 𝑥 ∈ No ( ( bday ‘ 𝑥 ) ⊆ 𝑎 → 𝑥 ∈ ( M ‘ 𝑎 ) ) ↔ ∀ 𝑥 ∈ No ( ( bday ‘ 𝑥 ) ⊆ 𝑏 → 𝑥 ∈ ( M ‘ 𝑏 ) ) ) ) |
7 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( bday ‘ 𝑥 ) = ( bday ‘ 𝑦 ) ) |
8 |
7
|
sseq1d |
⊢ ( 𝑥 = 𝑦 → ( ( bday ‘ 𝑥 ) ⊆ 𝑏 ↔ ( bday ‘ 𝑦 ) ⊆ 𝑏 ) ) |
9 |
|
eleq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ ( M ‘ 𝑏 ) ↔ 𝑦 ∈ ( M ‘ 𝑏 ) ) ) |
10 |
8 9
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( ( bday ‘ 𝑥 ) ⊆ 𝑏 → 𝑥 ∈ ( M ‘ 𝑏 ) ) ↔ ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) ) |
11 |
10
|
cbvralvw |
⊢ ( ∀ 𝑥 ∈ No ( ( bday ‘ 𝑥 ) ⊆ 𝑏 → 𝑥 ∈ ( M ‘ 𝑏 ) ) ↔ ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) |
12 |
6 11
|
bitrdi |
⊢ ( 𝑎 = 𝑏 → ( ∀ 𝑥 ∈ No ( ( bday ‘ 𝑥 ) ⊆ 𝑎 → 𝑥 ∈ ( M ‘ 𝑎 ) ) ↔ ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) ) |
13 |
|
sseq2 |
⊢ ( 𝑎 = 𝐴 → ( ( bday ‘ 𝑥 ) ⊆ 𝑎 ↔ ( bday ‘ 𝑥 ) ⊆ 𝐴 ) ) |
14 |
|
fveq2 |
⊢ ( 𝑎 = 𝐴 → ( M ‘ 𝑎 ) = ( M ‘ 𝐴 ) ) |
15 |
14
|
eleq2d |
⊢ ( 𝑎 = 𝐴 → ( 𝑥 ∈ ( M ‘ 𝑎 ) ↔ 𝑥 ∈ ( M ‘ 𝐴 ) ) ) |
16 |
13 15
|
imbi12d |
⊢ ( 𝑎 = 𝐴 → ( ( ( bday ‘ 𝑥 ) ⊆ 𝑎 → 𝑥 ∈ ( M ‘ 𝑎 ) ) ↔ ( ( bday ‘ 𝑥 ) ⊆ 𝐴 → 𝑥 ∈ ( M ‘ 𝐴 ) ) ) ) |
17 |
16
|
ralbidv |
⊢ ( 𝑎 = 𝐴 → ( ∀ 𝑥 ∈ No ( ( bday ‘ 𝑥 ) ⊆ 𝑎 → 𝑥 ∈ ( M ‘ 𝑎 ) ) ↔ ∀ 𝑥 ∈ No ( ( bday ‘ 𝑥 ) ⊆ 𝐴 → 𝑥 ∈ ( M ‘ 𝐴 ) ) ) ) |
18 |
|
bdayelon |
⊢ ( bday ‘ 𝑥 ) ∈ On |
19 |
|
onsseleq |
⊢ ( ( ( bday ‘ 𝑥 ) ∈ On ∧ 𝑎 ∈ On ) → ( ( bday ‘ 𝑥 ) ⊆ 𝑎 ↔ ( ( bday ‘ 𝑥 ) ∈ 𝑎 ∨ ( bday ‘ 𝑥 ) = 𝑎 ) ) ) |
20 |
18 19
|
mpan |
⊢ ( 𝑎 ∈ On → ( ( bday ‘ 𝑥 ) ⊆ 𝑎 ↔ ( ( bday ‘ 𝑥 ) ∈ 𝑎 ∨ ( bday ‘ 𝑥 ) = 𝑎 ) ) ) |
21 |
20
|
ad2antrr |
⊢ ( ( ( 𝑎 ∈ On ∧ ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) ∧ 𝑥 ∈ No ) → ( ( bday ‘ 𝑥 ) ⊆ 𝑎 ↔ ( ( bday ‘ 𝑥 ) ∈ 𝑎 ∨ ( bday ‘ 𝑥 ) = 𝑎 ) ) ) |
22 |
|
simpll |
⊢ ( ( ( 𝑎 ∈ On ∧ ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) ∧ 𝑥 ∈ No ) → 𝑎 ∈ On ) |
23 |
|
onelss |
⊢ ( 𝑎 ∈ On → ( ( bday ‘ 𝑥 ) ∈ 𝑎 → ( bday ‘ 𝑥 ) ⊆ 𝑎 ) ) |
24 |
23
|
ad2antrr |
⊢ ( ( ( 𝑎 ∈ On ∧ ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) ∧ 𝑥 ∈ No ) → ( ( bday ‘ 𝑥 ) ∈ 𝑎 → ( bday ‘ 𝑥 ) ⊆ 𝑎 ) ) |
25 |
24
|
imp |
⊢ ( ( ( ( 𝑎 ∈ On ∧ ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) ∧ 𝑥 ∈ No ) ∧ ( bday ‘ 𝑥 ) ∈ 𝑎 ) → ( bday ‘ 𝑥 ) ⊆ 𝑎 ) |
26 |
|
madess |
⊢ ( ( 𝑎 ∈ On ∧ ( bday ‘ 𝑥 ) ⊆ 𝑎 ) → ( M ‘ ( bday ‘ 𝑥 ) ) ⊆ ( M ‘ 𝑎 ) ) |
27 |
22 25 26
|
syl2an2r |
⊢ ( ( ( ( 𝑎 ∈ On ∧ ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) ∧ 𝑥 ∈ No ) ∧ ( bday ‘ 𝑥 ) ∈ 𝑎 ) → ( M ‘ ( bday ‘ 𝑥 ) ) ⊆ ( M ‘ 𝑎 ) ) |
28 |
|
ssid |
⊢ ( bday ‘ 𝑥 ) ⊆ ( bday ‘ 𝑥 ) |
29 |
|
simpr |
⊢ ( ( ( ( 𝑎 ∈ On ∧ ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) ∧ 𝑥 ∈ No ) ∧ ( bday ‘ 𝑥 ) ∈ 𝑎 ) → ( bday ‘ 𝑥 ) ∈ 𝑎 ) |
30 |
|
simplr |
⊢ ( ( ( ( 𝑎 ∈ On ∧ ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) ∧ 𝑥 ∈ No ) ∧ ( bday ‘ 𝑥 ) ∈ 𝑎 ) → 𝑥 ∈ No ) |
31 |
29 30
|
jca |
⊢ ( ( ( ( 𝑎 ∈ On ∧ ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) ∧ 𝑥 ∈ No ) ∧ ( bday ‘ 𝑥 ) ∈ 𝑎 ) → ( ( bday ‘ 𝑥 ) ∈ 𝑎 ∧ 𝑥 ∈ No ) ) |
32 |
|
simpllr |
⊢ ( ( ( ( 𝑎 ∈ On ∧ ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) ∧ 𝑥 ∈ No ) ∧ ( bday ‘ 𝑥 ) ∈ 𝑎 ) → ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) |
33 |
|
sseq2 |
⊢ ( 𝑏 = ( bday ‘ 𝑥 ) → ( ( bday ‘ 𝑦 ) ⊆ 𝑏 ↔ ( bday ‘ 𝑦 ) ⊆ ( bday ‘ 𝑥 ) ) ) |
34 |
|
fveq2 |
⊢ ( 𝑏 = ( bday ‘ 𝑥 ) → ( M ‘ 𝑏 ) = ( M ‘ ( bday ‘ 𝑥 ) ) ) |
35 |
34
|
eleq2d |
⊢ ( 𝑏 = ( bday ‘ 𝑥 ) → ( 𝑦 ∈ ( M ‘ 𝑏 ) ↔ 𝑦 ∈ ( M ‘ ( bday ‘ 𝑥 ) ) ) ) |
36 |
33 35
|
imbi12d |
⊢ ( 𝑏 = ( bday ‘ 𝑥 ) → ( ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ↔ ( ( bday ‘ 𝑦 ) ⊆ ( bday ‘ 𝑥 ) → 𝑦 ∈ ( M ‘ ( bday ‘ 𝑥 ) ) ) ) ) |
37 |
|
fveq2 |
⊢ ( 𝑦 = 𝑥 → ( bday ‘ 𝑦 ) = ( bday ‘ 𝑥 ) ) |
38 |
37
|
sseq1d |
⊢ ( 𝑦 = 𝑥 → ( ( bday ‘ 𝑦 ) ⊆ ( bday ‘ 𝑥 ) ↔ ( bday ‘ 𝑥 ) ⊆ ( bday ‘ 𝑥 ) ) ) |
39 |
|
eleq1 |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∈ ( M ‘ ( bday ‘ 𝑥 ) ) ↔ 𝑥 ∈ ( M ‘ ( bday ‘ 𝑥 ) ) ) ) |
40 |
38 39
|
imbi12d |
⊢ ( 𝑦 = 𝑥 → ( ( ( bday ‘ 𝑦 ) ⊆ ( bday ‘ 𝑥 ) → 𝑦 ∈ ( M ‘ ( bday ‘ 𝑥 ) ) ) ↔ ( ( bday ‘ 𝑥 ) ⊆ ( bday ‘ 𝑥 ) → 𝑥 ∈ ( M ‘ ( bday ‘ 𝑥 ) ) ) ) ) |
41 |
36 40
|
rspc2v |
⊢ ( ( ( bday ‘ 𝑥 ) ∈ 𝑎 ∧ 𝑥 ∈ No ) → ( ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) → ( ( bday ‘ 𝑥 ) ⊆ ( bday ‘ 𝑥 ) → 𝑥 ∈ ( M ‘ ( bday ‘ 𝑥 ) ) ) ) ) |
42 |
31 32 41
|
sylc |
⊢ ( ( ( ( 𝑎 ∈ On ∧ ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) ∧ 𝑥 ∈ No ) ∧ ( bday ‘ 𝑥 ) ∈ 𝑎 ) → ( ( bday ‘ 𝑥 ) ⊆ ( bday ‘ 𝑥 ) → 𝑥 ∈ ( M ‘ ( bday ‘ 𝑥 ) ) ) ) |
43 |
28 42
|
mpi |
⊢ ( ( ( ( 𝑎 ∈ On ∧ ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) ∧ 𝑥 ∈ No ) ∧ ( bday ‘ 𝑥 ) ∈ 𝑎 ) → 𝑥 ∈ ( M ‘ ( bday ‘ 𝑥 ) ) ) |
44 |
27 43
|
sseldd |
⊢ ( ( ( ( 𝑎 ∈ On ∧ ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) ∧ 𝑥 ∈ No ) ∧ ( bday ‘ 𝑥 ) ∈ 𝑎 ) → 𝑥 ∈ ( M ‘ 𝑎 ) ) |
45 |
44
|
ex |
⊢ ( ( ( 𝑎 ∈ On ∧ ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) ∧ 𝑥 ∈ No ) → ( ( bday ‘ 𝑥 ) ∈ 𝑎 → 𝑥 ∈ ( M ‘ 𝑎 ) ) ) |
46 |
|
madebdaylemlrcut |
⊢ ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑥 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑥 ∈ No ) → ( ( L ‘ 𝑥 ) |s ( R ‘ 𝑥 ) ) = 𝑥 ) |
47 |
18
|
a1i |
⊢ ( 𝑥 ∈ No → ( bday ‘ 𝑥 ) ∈ On ) |
48 |
|
lltropt |
⊢ ( 𝑥 ∈ No → ( L ‘ 𝑥 ) <<s ( R ‘ 𝑥 ) ) |
49 |
|
leftssold |
⊢ ( L ‘ 𝑥 ) ⊆ ( O ‘ ( bday ‘ 𝑥 ) ) |
50 |
49
|
a1i |
⊢ ( 𝑥 ∈ No → ( L ‘ 𝑥 ) ⊆ ( O ‘ ( bday ‘ 𝑥 ) ) ) |
51 |
|
rightssold |
⊢ ( R ‘ 𝑥 ) ⊆ ( O ‘ ( bday ‘ 𝑥 ) ) |
52 |
51
|
a1i |
⊢ ( 𝑥 ∈ No → ( R ‘ 𝑥 ) ⊆ ( O ‘ ( bday ‘ 𝑥 ) ) ) |
53 |
|
madecut |
⊢ ( ( ( ( bday ‘ 𝑥 ) ∈ On ∧ ( L ‘ 𝑥 ) <<s ( R ‘ 𝑥 ) ) ∧ ( ( L ‘ 𝑥 ) ⊆ ( O ‘ ( bday ‘ 𝑥 ) ) ∧ ( R ‘ 𝑥 ) ⊆ ( O ‘ ( bday ‘ 𝑥 ) ) ) ) → ( ( L ‘ 𝑥 ) |s ( R ‘ 𝑥 ) ) ∈ ( M ‘ ( bday ‘ 𝑥 ) ) ) |
54 |
47 48 50 52 53
|
syl22anc |
⊢ ( 𝑥 ∈ No → ( ( L ‘ 𝑥 ) |s ( R ‘ 𝑥 ) ) ∈ ( M ‘ ( bday ‘ 𝑥 ) ) ) |
55 |
54
|
adantl |
⊢ ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑥 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑥 ∈ No ) → ( ( L ‘ 𝑥 ) |s ( R ‘ 𝑥 ) ) ∈ ( M ‘ ( bday ‘ 𝑥 ) ) ) |
56 |
46 55
|
eqeltrrd |
⊢ ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑥 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑥 ∈ No ) → 𝑥 ∈ ( M ‘ ( bday ‘ 𝑥 ) ) ) |
57 |
|
raleq |
⊢ ( ( bday ‘ 𝑥 ) = 𝑎 → ( ∀ 𝑏 ∈ ( bday ‘ 𝑥 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ↔ ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) ) |
58 |
57
|
anbi1d |
⊢ ( ( bday ‘ 𝑥 ) = 𝑎 → ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑥 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑥 ∈ No ) ↔ ( ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑥 ∈ No ) ) ) |
59 |
|
fveq2 |
⊢ ( ( bday ‘ 𝑥 ) = 𝑎 → ( M ‘ ( bday ‘ 𝑥 ) ) = ( M ‘ 𝑎 ) ) |
60 |
59
|
eleq2d |
⊢ ( ( bday ‘ 𝑥 ) = 𝑎 → ( 𝑥 ∈ ( M ‘ ( bday ‘ 𝑥 ) ) ↔ 𝑥 ∈ ( M ‘ 𝑎 ) ) ) |
61 |
58 60
|
imbi12d |
⊢ ( ( bday ‘ 𝑥 ) = 𝑎 → ( ( ( ∀ 𝑏 ∈ ( bday ‘ 𝑥 ) ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑥 ∈ No ) → 𝑥 ∈ ( M ‘ ( bday ‘ 𝑥 ) ) ) ↔ ( ( ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑥 ∈ No ) → 𝑥 ∈ ( M ‘ 𝑎 ) ) ) ) |
62 |
56 61
|
mpbii |
⊢ ( ( bday ‘ 𝑥 ) = 𝑎 → ( ( ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑥 ∈ No ) → 𝑥 ∈ ( M ‘ 𝑎 ) ) ) |
63 |
62
|
com12 |
⊢ ( ( ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ∧ 𝑥 ∈ No ) → ( ( bday ‘ 𝑥 ) = 𝑎 → 𝑥 ∈ ( M ‘ 𝑎 ) ) ) |
64 |
63
|
adantll |
⊢ ( ( ( 𝑎 ∈ On ∧ ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) ∧ 𝑥 ∈ No ) → ( ( bday ‘ 𝑥 ) = 𝑎 → 𝑥 ∈ ( M ‘ 𝑎 ) ) ) |
65 |
45 64
|
jaod |
⊢ ( ( ( 𝑎 ∈ On ∧ ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) ∧ 𝑥 ∈ No ) → ( ( ( bday ‘ 𝑥 ) ∈ 𝑎 ∨ ( bday ‘ 𝑥 ) = 𝑎 ) → 𝑥 ∈ ( M ‘ 𝑎 ) ) ) |
66 |
21 65
|
sylbid |
⊢ ( ( ( 𝑎 ∈ On ∧ ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) ∧ 𝑥 ∈ No ) → ( ( bday ‘ 𝑥 ) ⊆ 𝑎 → 𝑥 ∈ ( M ‘ 𝑎 ) ) ) |
67 |
66
|
ralrimiva |
⊢ ( ( 𝑎 ∈ On ∧ ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) ) → ∀ 𝑥 ∈ No ( ( bday ‘ 𝑥 ) ⊆ 𝑎 → 𝑥 ∈ ( M ‘ 𝑎 ) ) ) |
68 |
67
|
ex |
⊢ ( 𝑎 ∈ On → ( ∀ 𝑏 ∈ 𝑎 ∀ 𝑦 ∈ No ( ( bday ‘ 𝑦 ) ⊆ 𝑏 → 𝑦 ∈ ( M ‘ 𝑏 ) ) → ∀ 𝑥 ∈ No ( ( bday ‘ 𝑥 ) ⊆ 𝑎 → 𝑥 ∈ ( M ‘ 𝑎 ) ) ) ) |
69 |
12 17 68
|
tfis3 |
⊢ ( 𝐴 ∈ On → ∀ 𝑥 ∈ No ( ( bday ‘ 𝑥 ) ⊆ 𝐴 → 𝑥 ∈ ( M ‘ 𝐴 ) ) ) |
70 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( bday ‘ 𝑥 ) = ( bday ‘ 𝑋 ) ) |
71 |
70
|
sseq1d |
⊢ ( 𝑥 = 𝑋 → ( ( bday ‘ 𝑥 ) ⊆ 𝐴 ↔ ( bday ‘ 𝑋 ) ⊆ 𝐴 ) ) |
72 |
|
eleq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 ∈ ( M ‘ 𝐴 ) ↔ 𝑋 ∈ ( M ‘ 𝐴 ) ) ) |
73 |
71 72
|
imbi12d |
⊢ ( 𝑥 = 𝑋 → ( ( ( bday ‘ 𝑥 ) ⊆ 𝐴 → 𝑥 ∈ ( M ‘ 𝐴 ) ) ↔ ( ( bday ‘ 𝑋 ) ⊆ 𝐴 → 𝑋 ∈ ( M ‘ 𝐴 ) ) ) ) |
74 |
73
|
rspccva |
⊢ ( ( ∀ 𝑥 ∈ No ( ( bday ‘ 𝑥 ) ⊆ 𝐴 → 𝑥 ∈ ( M ‘ 𝐴 ) ) ∧ 𝑋 ∈ No ) → ( ( bday ‘ 𝑋 ) ⊆ 𝐴 → 𝑋 ∈ ( M ‘ 𝐴 ) ) ) |
75 |
69 74
|
sylan |
⊢ ( ( 𝐴 ∈ On ∧ 𝑋 ∈ No ) → ( ( bday ‘ 𝑋 ) ⊆ 𝐴 → 𝑋 ∈ ( M ‘ 𝐴 ) ) ) |
76 |
1 75
|
impbid2 |
⊢ ( ( 𝐴 ∈ On ∧ 𝑋 ∈ No ) → ( 𝑋 ∈ ( M ‘ 𝐴 ) ↔ ( bday ‘ 𝑋 ) ⊆ 𝐴 ) ) |