Step |
Hyp |
Ref |
Expression |
1 |
|
frrlem4.1 |
⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } |
2 |
1
|
frrlem2 |
⊢ ( 𝑔 ∈ 𝐵 → Fun 𝑔 ) |
3 |
2
|
funfnd |
⊢ ( 𝑔 ∈ 𝐵 → 𝑔 Fn dom 𝑔 ) |
4 |
|
fnresin1 |
⊢ ( 𝑔 Fn dom 𝑔 → ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) Fn ( dom 𝑔 ∩ dom ℎ ) ) |
5 |
3 4
|
syl |
⊢ ( 𝑔 ∈ 𝐵 → ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) Fn ( dom 𝑔 ∩ dom ℎ ) ) |
6 |
5
|
adantr |
⊢ ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) → ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) Fn ( dom 𝑔 ∩ dom ℎ ) ) |
7 |
1
|
frrlem1 |
⊢ 𝐵 = { 𝑔 ∣ ∃ 𝑏 ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ∧ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) } |
8 |
7
|
abeq2i |
⊢ ( 𝑔 ∈ 𝐵 ↔ ∃ 𝑏 ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ∧ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) |
9 |
|
fndm |
⊢ ( 𝑔 Fn 𝑏 → dom 𝑔 = 𝑏 ) |
10 |
9
|
adantr |
⊢ ( ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ) → dom 𝑔 = 𝑏 ) |
11 |
10
|
raleqdv |
⊢ ( ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ) → ( ∀ 𝑎 ∈ dom 𝑔 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ↔ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) |
12 |
11
|
biimp3ar |
⊢ ( ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ∧ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) → ∀ 𝑎 ∈ dom 𝑔 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) |
13 |
|
rsp |
⊢ ( ∀ 𝑎 ∈ dom 𝑔 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) → ( 𝑎 ∈ dom 𝑔 → ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) |
14 |
12 13
|
syl |
⊢ ( ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ∧ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) → ( 𝑎 ∈ dom 𝑔 → ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) |
15 |
14
|
exlimiv |
⊢ ( ∃ 𝑏 ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ∧ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) → ( 𝑎 ∈ dom 𝑔 → ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) |
16 |
8 15
|
sylbi |
⊢ ( 𝑔 ∈ 𝐵 → ( 𝑎 ∈ dom 𝑔 → ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) |
17 |
|
elinel1 |
⊢ ( 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) → 𝑎 ∈ dom 𝑔 ) |
18 |
16 17
|
impel |
⊢ ( ( 𝑔 ∈ 𝐵 ∧ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ) → ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) |
19 |
18
|
adantlr |
⊢ ( ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) ∧ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ) → ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) |
20 |
|
simpr |
⊢ ( ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) ∧ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ) → 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ) |
21 |
20
|
fvresd |
⊢ ( ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) ∧ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ) → ( ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) ‘ 𝑎 ) = ( 𝑔 ‘ 𝑎 ) ) |
22 |
|
resres |
⊢ ( ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) ↾ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) = ( 𝑔 ↾ ( ( dom 𝑔 ∩ dom ℎ ) ∩ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) ) |
23 |
|
predss |
⊢ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ⊆ ( dom 𝑔 ∩ dom ℎ ) |
24 |
|
sseqin2 |
⊢ ( Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ⊆ ( dom 𝑔 ∩ dom ℎ ) ↔ ( ( dom 𝑔 ∩ dom ℎ ) ∩ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) = Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) |
25 |
23 24
|
mpbi |
⊢ ( ( dom 𝑔 ∩ dom ℎ ) ∩ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) = Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) |
26 |
1
|
frrlem1 |
⊢ 𝐵 = { ℎ ∣ ∃ 𝑐 ( ℎ Fn 𝑐 ∧ ( 𝑐 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ∧ ∀ 𝑎 ∈ 𝑐 ( ℎ ‘ 𝑎 ) = ( 𝑎 𝐺 ( ℎ ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) } |
27 |
26
|
abeq2i |
⊢ ( ℎ ∈ 𝐵 ↔ ∃ 𝑐 ( ℎ Fn 𝑐 ∧ ( 𝑐 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ∧ ∀ 𝑎 ∈ 𝑐 ( ℎ ‘ 𝑎 ) = ( 𝑎 𝐺 ( ℎ ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) |
28 |
|
exdistrv |
⊢ ( ∃ 𝑏 ∃ 𝑐 ( ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ∧ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ∧ ( ℎ Fn 𝑐 ∧ ( 𝑐 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ∧ ∀ 𝑎 ∈ 𝑐 ( ℎ ‘ 𝑎 ) = ( 𝑎 𝐺 ( ℎ ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) ↔ ( ∃ 𝑏 ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ∧ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ∧ ∃ 𝑐 ( ℎ Fn 𝑐 ∧ ( 𝑐 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ∧ ∀ 𝑎 ∈ 𝑐 ( ℎ ‘ 𝑎 ) = ( 𝑎 𝐺 ( ℎ ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) ) |
29 |
|
inss1 |
⊢ ( 𝑏 ∩ 𝑐 ) ⊆ 𝑏 |
30 |
|
simpl2l |
⊢ ( ( ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ∧ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ∧ ( ℎ Fn 𝑐 ∧ ( 𝑐 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ∧ ∀ 𝑎 ∈ 𝑐 ( ℎ ‘ 𝑎 ) = ( 𝑎 𝐺 ( ℎ ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) → 𝑏 ⊆ 𝐴 ) |
31 |
29 30
|
sstrid |
⊢ ( ( ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ∧ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ∧ ( ℎ Fn 𝑐 ∧ ( 𝑐 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ∧ ∀ 𝑎 ∈ 𝑐 ( ℎ ‘ 𝑎 ) = ( 𝑎 𝐺 ( ℎ ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) → ( 𝑏 ∩ 𝑐 ) ⊆ 𝐴 ) |
32 |
|
simp2r |
⊢ ( ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ∧ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) → ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) |
33 |
|
simp2r |
⊢ ( ( ℎ Fn 𝑐 ∧ ( 𝑐 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ∧ ∀ 𝑎 ∈ 𝑐 ( ℎ ‘ 𝑎 ) = ( 𝑎 𝐺 ( ℎ ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) → ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) |
34 |
|
nfra1 |
⊢ Ⅎ 𝑎 ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 |
35 |
|
nfra1 |
⊢ Ⅎ 𝑎 ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 |
36 |
34 35
|
nfan |
⊢ Ⅎ 𝑎 ( ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) |
37 |
|
elinel1 |
⊢ ( 𝑎 ∈ ( 𝑏 ∩ 𝑐 ) → 𝑎 ∈ 𝑏 ) |
38 |
|
rsp |
⊢ ( ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 → ( 𝑎 ∈ 𝑏 → Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ) |
39 |
37 38
|
syl5com |
⊢ ( 𝑎 ∈ ( 𝑏 ∩ 𝑐 ) → ( ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 → Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ) |
40 |
|
elinel2 |
⊢ ( 𝑎 ∈ ( 𝑏 ∩ 𝑐 ) → 𝑎 ∈ 𝑐 ) |
41 |
|
rsp |
⊢ ( ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 → ( 𝑎 ∈ 𝑐 → Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ) |
42 |
40 41
|
syl5com |
⊢ ( 𝑎 ∈ ( 𝑏 ∩ 𝑐 ) → ( ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 → Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ) |
43 |
39 42
|
anim12d |
⊢ ( 𝑎 ∈ ( 𝑏 ∩ 𝑐 ) → ( ( ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) → ( Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ∧ Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ) ) |
44 |
|
ssin |
⊢ ( ( Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ∧ Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ↔ Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ ( 𝑏 ∩ 𝑐 ) ) |
45 |
44
|
biimpi |
⊢ ( ( Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ∧ Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) → Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ ( 𝑏 ∩ 𝑐 ) ) |
46 |
43 45
|
syl6com |
⊢ ( ( ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) → ( 𝑎 ∈ ( 𝑏 ∩ 𝑐 ) → Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ ( 𝑏 ∩ 𝑐 ) ) ) |
47 |
36 46
|
ralrimi |
⊢ ( ( ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) → ∀ 𝑎 ∈ ( 𝑏 ∩ 𝑐 ) Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ ( 𝑏 ∩ 𝑐 ) ) |
48 |
32 33 47
|
syl2an |
⊢ ( ( ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ∧ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ∧ ( ℎ Fn 𝑐 ∧ ( 𝑐 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ∧ ∀ 𝑎 ∈ 𝑐 ( ℎ ‘ 𝑎 ) = ( 𝑎 𝐺 ( ℎ ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) → ∀ 𝑎 ∈ ( 𝑏 ∩ 𝑐 ) Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ ( 𝑏 ∩ 𝑐 ) ) |
49 |
|
simpl1 |
⊢ ( ( ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ∧ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ∧ ( ℎ Fn 𝑐 ∧ ( 𝑐 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ∧ ∀ 𝑎 ∈ 𝑐 ( ℎ ‘ 𝑎 ) = ( 𝑎 𝐺 ( ℎ ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) → 𝑔 Fn 𝑏 ) |
50 |
49
|
fndmd |
⊢ ( ( ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ∧ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ∧ ( ℎ Fn 𝑐 ∧ ( 𝑐 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ∧ ∀ 𝑎 ∈ 𝑐 ( ℎ ‘ 𝑎 ) = ( 𝑎 𝐺 ( ℎ ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) → dom 𝑔 = 𝑏 ) |
51 |
|
simpr1 |
⊢ ( ( ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ∧ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ∧ ( ℎ Fn 𝑐 ∧ ( 𝑐 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ∧ ∀ 𝑎 ∈ 𝑐 ( ℎ ‘ 𝑎 ) = ( 𝑎 𝐺 ( ℎ ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) → ℎ Fn 𝑐 ) |
52 |
51
|
fndmd |
⊢ ( ( ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ∧ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ∧ ( ℎ Fn 𝑐 ∧ ( 𝑐 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ∧ ∀ 𝑎 ∈ 𝑐 ( ℎ ‘ 𝑎 ) = ( 𝑎 𝐺 ( ℎ ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) → dom ℎ = 𝑐 ) |
53 |
|
ineq12 |
⊢ ( ( dom 𝑔 = 𝑏 ∧ dom ℎ = 𝑐 ) → ( dom 𝑔 ∩ dom ℎ ) = ( 𝑏 ∩ 𝑐 ) ) |
54 |
53
|
sseq1d |
⊢ ( ( dom 𝑔 = 𝑏 ∧ dom ℎ = 𝑐 ) → ( ( dom 𝑔 ∩ dom ℎ ) ⊆ 𝐴 ↔ ( 𝑏 ∩ 𝑐 ) ⊆ 𝐴 ) ) |
55 |
53
|
sseq2d |
⊢ ( ( dom 𝑔 = 𝑏 ∧ dom ℎ = 𝑐 ) → ( Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ ( dom 𝑔 ∩ dom ℎ ) ↔ Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ ( 𝑏 ∩ 𝑐 ) ) ) |
56 |
53 55
|
raleqbidv |
⊢ ( ( dom 𝑔 = 𝑏 ∧ dom ℎ = 𝑐 ) → ( ∀ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ ( dom 𝑔 ∩ dom ℎ ) ↔ ∀ 𝑎 ∈ ( 𝑏 ∩ 𝑐 ) Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ ( 𝑏 ∩ 𝑐 ) ) ) |
57 |
54 56
|
anbi12d |
⊢ ( ( dom 𝑔 = 𝑏 ∧ dom ℎ = 𝑐 ) → ( ( ( dom 𝑔 ∩ dom ℎ ) ⊆ 𝐴 ∧ ∀ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ ( dom 𝑔 ∩ dom ℎ ) ) ↔ ( ( 𝑏 ∩ 𝑐 ) ⊆ 𝐴 ∧ ∀ 𝑎 ∈ ( 𝑏 ∩ 𝑐 ) Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ ( 𝑏 ∩ 𝑐 ) ) ) ) |
58 |
50 52 57
|
syl2anc |
⊢ ( ( ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ∧ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ∧ ( ℎ Fn 𝑐 ∧ ( 𝑐 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ∧ ∀ 𝑎 ∈ 𝑐 ( ℎ ‘ 𝑎 ) = ( 𝑎 𝐺 ( ℎ ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) → ( ( ( dom 𝑔 ∩ dom ℎ ) ⊆ 𝐴 ∧ ∀ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ ( dom 𝑔 ∩ dom ℎ ) ) ↔ ( ( 𝑏 ∩ 𝑐 ) ⊆ 𝐴 ∧ ∀ 𝑎 ∈ ( 𝑏 ∩ 𝑐 ) Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ ( 𝑏 ∩ 𝑐 ) ) ) ) |
59 |
31 48 58
|
mpbir2and |
⊢ ( ( ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ∧ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ∧ ( ℎ Fn 𝑐 ∧ ( 𝑐 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ∧ ∀ 𝑎 ∈ 𝑐 ( ℎ ‘ 𝑎 ) = ( 𝑎 𝐺 ( ℎ ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) → ( ( dom 𝑔 ∩ dom ℎ ) ⊆ 𝐴 ∧ ∀ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ ( dom 𝑔 ∩ dom ℎ ) ) ) |
60 |
59
|
exlimivv |
⊢ ( ∃ 𝑏 ∃ 𝑐 ( ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ∧ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ∧ ( ℎ Fn 𝑐 ∧ ( 𝑐 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ∧ ∀ 𝑎 ∈ 𝑐 ( ℎ ‘ 𝑎 ) = ( 𝑎 𝐺 ( ℎ ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) → ( ( dom 𝑔 ∩ dom ℎ ) ⊆ 𝐴 ∧ ∀ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ ( dom 𝑔 ∩ dom ℎ ) ) ) |
61 |
28 60
|
sylbir |
⊢ ( ( ∃ 𝑏 ( 𝑔 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑏 ) ∧ ∀ 𝑎 ∈ 𝑏 ( 𝑔 ‘ 𝑎 ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ∧ ∃ 𝑐 ( ℎ Fn 𝑐 ∧ ( 𝑐 ⊆ 𝐴 ∧ ∀ 𝑎 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ 𝑐 ) ∧ ∀ 𝑎 ∈ 𝑐 ( ℎ ‘ 𝑎 ) = ( 𝑎 𝐺 ( ℎ ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) ) → ( ( dom 𝑔 ∩ dom ℎ ) ⊆ 𝐴 ∧ ∀ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ ( dom 𝑔 ∩ dom ℎ ) ) ) |
62 |
8 27 61
|
syl2anb |
⊢ ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) → ( ( dom 𝑔 ∩ dom ℎ ) ⊆ 𝐴 ∧ ∀ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ ( dom 𝑔 ∩ dom ℎ ) ) ) |
63 |
62
|
adantr |
⊢ ( ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) ∧ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ) → ( ( dom 𝑔 ∩ dom ℎ ) ⊆ 𝐴 ∧ ∀ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ ( dom 𝑔 ∩ dom ℎ ) ) ) |
64 |
|
preddowncl |
⊢ ( ( ( dom 𝑔 ∩ dom ℎ ) ⊆ 𝐴 ∧ ∀ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ ( dom 𝑔 ∩ dom ℎ ) ) → ( 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) → Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) = Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) |
65 |
63 20 64
|
sylc |
⊢ ( ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) ∧ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ) → Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) = Pred ( 𝑅 , 𝐴 , 𝑎 ) ) |
66 |
25 65
|
eqtrid |
⊢ ( ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) ∧ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ) → ( ( dom 𝑔 ∩ dom ℎ ) ∩ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) = Pred ( 𝑅 , 𝐴 , 𝑎 ) ) |
67 |
66
|
reseq2d |
⊢ ( ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) ∧ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ) → ( 𝑔 ↾ ( ( dom 𝑔 ∩ dom ℎ ) ∩ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) ) = ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) |
68 |
22 67
|
eqtrid |
⊢ ( ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) ∧ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ) → ( ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) ↾ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) = ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) |
69 |
68
|
oveq2d |
⊢ ( ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) ∧ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ) → ( 𝑎 𝐺 ( ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) ↾ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) ) = ( 𝑎 𝐺 ( 𝑔 ↾ Pred ( 𝑅 , 𝐴 , 𝑎 ) ) ) ) |
70 |
19 21 69
|
3eqtr4d |
⊢ ( ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) ∧ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ) → ( ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) ‘ 𝑎 ) = ( 𝑎 𝐺 ( ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) ↾ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) ) ) |
71 |
70
|
ralrimiva |
⊢ ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) → ∀ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ( ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) ‘ 𝑎 ) = ( 𝑎 𝐺 ( ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) ↾ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) ) ) |
72 |
6 71
|
jca |
⊢ ( ( 𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) → ( ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) Fn ( dom 𝑔 ∩ dom ℎ ) ∧ ∀ 𝑎 ∈ ( dom 𝑔 ∩ dom ℎ ) ( ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) ‘ 𝑎 ) = ( 𝑎 𝐺 ( ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) ↾ Pred ( 𝑅 , ( dom 𝑔 ∩ dom ℎ ) , 𝑎 ) ) ) ) ) |