| Step |
Hyp |
Ref |
Expression |
| 1 |
|
cfsmolem.2 |
⊢ 𝐹 = ( 𝑧 ∈ V ↦ ( ( 𝑔 ‘ dom 𝑧 ) ∪ ∪ 𝑡 ∈ dom 𝑧 suc ( 𝑧 ‘ 𝑡 ) ) ) |
| 2 |
|
cfsmolem.3 |
⊢ 𝐺 = ( recs ( 𝐹 ) ↾ ( cf ‘ 𝐴 ) ) |
| 3 |
|
cff1 |
⊢ ( 𝐴 ∈ On → ∃ 𝑔 ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝑔 ‘ 𝑤 ) ) ) |
| 4 |
|
cfon |
⊢ ( cf ‘ 𝐴 ) ∈ On |
| 5 |
4
|
oneli |
⊢ ( 𝑥 ∈ ( cf ‘ 𝐴 ) → 𝑥 ∈ On ) |
| 6 |
5
|
3ad2ant3 |
⊢ ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → 𝑥 ∈ On ) |
| 7 |
|
eleq1w |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ ( cf ‘ 𝐴 ) ↔ 𝑦 ∈ ( cf ‘ 𝐴 ) ) ) |
| 8 |
7
|
3anbi3d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) ↔ ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑦 ∈ ( cf ‘ 𝐴 ) ) ) ) |
| 9 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐺 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑦 ) ) |
| 10 |
9
|
eleq1d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐺 ‘ 𝑥 ) ∈ 𝐴 ↔ ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ) |
| 11 |
8 10
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( 𝐺 ‘ 𝑥 ) ∈ 𝐴 ) ↔ ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑦 ∈ ( cf ‘ 𝐴 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ) ) |
| 12 |
|
simpl1 |
⊢ ( ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑥 ) → 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ) |
| 13 |
|
simpl2 |
⊢ ( ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑥 ) → 𝐴 ∈ On ) |
| 14 |
|
ontr1 |
⊢ ( ( cf ‘ 𝐴 ) ∈ On → ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → 𝑦 ∈ ( cf ‘ 𝐴 ) ) ) |
| 15 |
4 14
|
ax-mp |
⊢ ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → 𝑦 ∈ ( cf ‘ 𝐴 ) ) |
| 16 |
15
|
ancoms |
⊢ ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ∈ ( cf ‘ 𝐴 ) ) |
| 17 |
16
|
3ad2antl3 |
⊢ ( ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ∈ ( cf ‘ 𝐴 ) ) |
| 18 |
|
pm2.27 |
⊢ ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑦 ∈ ( cf ‘ 𝐴 ) ) → ( ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑦 ∈ ( cf ‘ 𝐴 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ) |
| 19 |
12 13 17 18
|
syl3anc |
⊢ ( ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑥 ) → ( ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑦 ∈ ( cf ‘ 𝐴 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ) |
| 20 |
19
|
ralimdva |
⊢ ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( ∀ 𝑦 ∈ 𝑥 ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑦 ∈ ( cf ‘ 𝐴 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ) |
| 21 |
2
|
fveq1i |
⊢ ( 𝐺 ‘ 𝑥 ) = ( ( recs ( 𝐹 ) ↾ ( cf ‘ 𝐴 ) ) ‘ 𝑥 ) |
| 22 |
|
fvres |
⊢ ( 𝑥 ∈ ( cf ‘ 𝐴 ) → ( ( recs ( 𝐹 ) ↾ ( cf ‘ 𝐴 ) ) ‘ 𝑥 ) = ( recs ( 𝐹 ) ‘ 𝑥 ) ) |
| 23 |
21 22
|
eqtrid |
⊢ ( 𝑥 ∈ ( cf ‘ 𝐴 ) → ( 𝐺 ‘ 𝑥 ) = ( recs ( 𝐹 ) ‘ 𝑥 ) ) |
| 24 |
|
recsval |
⊢ ( 𝑥 ∈ On → ( recs ( 𝐹 ) ‘ 𝑥 ) = ( 𝐹 ‘ ( recs ( 𝐹 ) ↾ 𝑥 ) ) ) |
| 25 |
|
recsfnon |
⊢ recs ( 𝐹 ) Fn On |
| 26 |
|
fnfun |
⊢ ( recs ( 𝐹 ) Fn On → Fun recs ( 𝐹 ) ) |
| 27 |
25 26
|
ax-mp |
⊢ Fun recs ( 𝐹 ) |
| 28 |
|
vex |
⊢ 𝑥 ∈ V |
| 29 |
|
resfunexg |
⊢ ( ( Fun recs ( 𝐹 ) ∧ 𝑥 ∈ V ) → ( recs ( 𝐹 ) ↾ 𝑥 ) ∈ V ) |
| 30 |
27 28 29
|
mp2an |
⊢ ( recs ( 𝐹 ) ↾ 𝑥 ) ∈ V |
| 31 |
|
dmeq |
⊢ ( 𝑧 = ( recs ( 𝐹 ) ↾ 𝑥 ) → dom 𝑧 = dom ( recs ( 𝐹 ) ↾ 𝑥 ) ) |
| 32 |
31
|
fveq2d |
⊢ ( 𝑧 = ( recs ( 𝐹 ) ↾ 𝑥 ) → ( 𝑔 ‘ dom 𝑧 ) = ( 𝑔 ‘ dom ( recs ( 𝐹 ) ↾ 𝑥 ) ) ) |
| 33 |
|
fveq1 |
⊢ ( 𝑧 = ( recs ( 𝐹 ) ↾ 𝑥 ) → ( 𝑧 ‘ 𝑡 ) = ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) ) |
| 34 |
|
suceq |
⊢ ( ( 𝑧 ‘ 𝑡 ) = ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) → suc ( 𝑧 ‘ 𝑡 ) = suc ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) ) |
| 35 |
33 34
|
syl |
⊢ ( 𝑧 = ( recs ( 𝐹 ) ↾ 𝑥 ) → suc ( 𝑧 ‘ 𝑡 ) = suc ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) ) |
| 36 |
31 35
|
iuneq12d |
⊢ ( 𝑧 = ( recs ( 𝐹 ) ↾ 𝑥 ) → ∪ 𝑡 ∈ dom 𝑧 suc ( 𝑧 ‘ 𝑡 ) = ∪ 𝑡 ∈ dom ( recs ( 𝐹 ) ↾ 𝑥 ) suc ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) ) |
| 37 |
32 36
|
uneq12d |
⊢ ( 𝑧 = ( recs ( 𝐹 ) ↾ 𝑥 ) → ( ( 𝑔 ‘ dom 𝑧 ) ∪ ∪ 𝑡 ∈ dom 𝑧 suc ( 𝑧 ‘ 𝑡 ) ) = ( ( 𝑔 ‘ dom ( recs ( 𝐹 ) ↾ 𝑥 ) ) ∪ ∪ 𝑡 ∈ dom ( recs ( 𝐹 ) ↾ 𝑥 ) suc ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) ) ) |
| 38 |
|
fvex |
⊢ ( 𝑔 ‘ dom ( recs ( 𝐹 ) ↾ 𝑥 ) ) ∈ V |
| 39 |
30
|
dmex |
⊢ dom ( recs ( 𝐹 ) ↾ 𝑥 ) ∈ V |
| 40 |
|
fvex |
⊢ ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) ∈ V |
| 41 |
40
|
sucex |
⊢ suc ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) ∈ V |
| 42 |
39 41
|
iunex |
⊢ ∪ 𝑡 ∈ dom ( recs ( 𝐹 ) ↾ 𝑥 ) suc ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) ∈ V |
| 43 |
38 42
|
unex |
⊢ ( ( 𝑔 ‘ dom ( recs ( 𝐹 ) ↾ 𝑥 ) ) ∪ ∪ 𝑡 ∈ dom ( recs ( 𝐹 ) ↾ 𝑥 ) suc ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) ) ∈ V |
| 44 |
37 1 43
|
fvmpt |
⊢ ( ( recs ( 𝐹 ) ↾ 𝑥 ) ∈ V → ( 𝐹 ‘ ( recs ( 𝐹 ) ↾ 𝑥 ) ) = ( ( 𝑔 ‘ dom ( recs ( 𝐹 ) ↾ 𝑥 ) ) ∪ ∪ 𝑡 ∈ dom ( recs ( 𝐹 ) ↾ 𝑥 ) suc ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) ) ) |
| 45 |
30 44
|
ax-mp |
⊢ ( 𝐹 ‘ ( recs ( 𝐹 ) ↾ 𝑥 ) ) = ( ( 𝑔 ‘ dom ( recs ( 𝐹 ) ↾ 𝑥 ) ) ∪ ∪ 𝑡 ∈ dom ( recs ( 𝐹 ) ↾ 𝑥 ) suc ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) ) |
| 46 |
24 45
|
eqtrdi |
⊢ ( 𝑥 ∈ On → ( recs ( 𝐹 ) ‘ 𝑥 ) = ( ( 𝑔 ‘ dom ( recs ( 𝐹 ) ↾ 𝑥 ) ) ∪ ∪ 𝑡 ∈ dom ( recs ( 𝐹 ) ↾ 𝑥 ) suc ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) ) ) |
| 47 |
|
onss |
⊢ ( 𝑥 ∈ On → 𝑥 ⊆ On ) |
| 48 |
|
fnssres |
⊢ ( ( recs ( 𝐹 ) Fn On ∧ 𝑥 ⊆ On ) → ( recs ( 𝐹 ) ↾ 𝑥 ) Fn 𝑥 ) |
| 49 |
25 47 48
|
sylancr |
⊢ ( 𝑥 ∈ On → ( recs ( 𝐹 ) ↾ 𝑥 ) Fn 𝑥 ) |
| 50 |
|
fndm |
⊢ ( ( recs ( 𝐹 ) ↾ 𝑥 ) Fn 𝑥 → dom ( recs ( 𝐹 ) ↾ 𝑥 ) = 𝑥 ) |
| 51 |
|
fveq2 |
⊢ ( dom ( recs ( 𝐹 ) ↾ 𝑥 ) = 𝑥 → ( 𝑔 ‘ dom ( recs ( 𝐹 ) ↾ 𝑥 ) ) = ( 𝑔 ‘ 𝑥 ) ) |
| 52 |
|
iuneq1 |
⊢ ( dom ( recs ( 𝐹 ) ↾ 𝑥 ) = 𝑥 → ∪ 𝑡 ∈ dom ( recs ( 𝐹 ) ↾ 𝑥 ) suc ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) = ∪ 𝑡 ∈ 𝑥 suc ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) ) |
| 53 |
|
fvres |
⊢ ( 𝑡 ∈ 𝑥 → ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) = ( recs ( 𝐹 ) ‘ 𝑡 ) ) |
| 54 |
|
suceq |
⊢ ( ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) = ( recs ( 𝐹 ) ‘ 𝑡 ) → suc ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) = suc ( recs ( 𝐹 ) ‘ 𝑡 ) ) |
| 55 |
53 54
|
syl |
⊢ ( 𝑡 ∈ 𝑥 → suc ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) = suc ( recs ( 𝐹 ) ‘ 𝑡 ) ) |
| 56 |
55
|
iuneq2i |
⊢ ∪ 𝑡 ∈ 𝑥 suc ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) = ∪ 𝑡 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑡 ) |
| 57 |
|
fveq2 |
⊢ ( 𝑦 = 𝑡 → ( recs ( 𝐹 ) ‘ 𝑦 ) = ( recs ( 𝐹 ) ‘ 𝑡 ) ) |
| 58 |
|
suceq |
⊢ ( ( recs ( 𝐹 ) ‘ 𝑦 ) = ( recs ( 𝐹 ) ‘ 𝑡 ) → suc ( recs ( 𝐹 ) ‘ 𝑦 ) = suc ( recs ( 𝐹 ) ‘ 𝑡 ) ) |
| 59 |
57 58
|
syl |
⊢ ( 𝑦 = 𝑡 → suc ( recs ( 𝐹 ) ‘ 𝑦 ) = suc ( recs ( 𝐹 ) ‘ 𝑡 ) ) |
| 60 |
59
|
cbviunv |
⊢ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = ∪ 𝑡 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑡 ) |
| 61 |
56 60
|
eqtr4i |
⊢ ∪ 𝑡 ∈ 𝑥 suc ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) = ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) |
| 62 |
52 61
|
eqtrdi |
⊢ ( dom ( recs ( 𝐹 ) ↾ 𝑥 ) = 𝑥 → ∪ 𝑡 ∈ dom ( recs ( 𝐹 ) ↾ 𝑥 ) suc ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) = ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 63 |
51 62
|
uneq12d |
⊢ ( dom ( recs ( 𝐹 ) ↾ 𝑥 ) = 𝑥 → ( ( 𝑔 ‘ dom ( recs ( 𝐹 ) ↾ 𝑥 ) ) ∪ ∪ 𝑡 ∈ dom ( recs ( 𝐹 ) ↾ 𝑥 ) suc ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) ) = ( ( 𝑔 ‘ 𝑥 ) ∪ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 64 |
49 50 63
|
3syl |
⊢ ( 𝑥 ∈ On → ( ( 𝑔 ‘ dom ( recs ( 𝐹 ) ↾ 𝑥 ) ) ∪ ∪ 𝑡 ∈ dom ( recs ( 𝐹 ) ↾ 𝑥 ) suc ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑡 ) ) = ( ( 𝑔 ‘ 𝑥 ) ∪ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 65 |
46 64
|
eqtrd |
⊢ ( 𝑥 ∈ On → ( recs ( 𝐹 ) ‘ 𝑥 ) = ( ( 𝑔 ‘ 𝑥 ) ∪ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 66 |
5 65
|
syl |
⊢ ( 𝑥 ∈ ( cf ‘ 𝐴 ) → ( recs ( 𝐹 ) ‘ 𝑥 ) = ( ( 𝑔 ‘ 𝑥 ) ∪ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 67 |
23 66
|
eqtrd |
⊢ ( 𝑥 ∈ ( cf ‘ 𝐴 ) → ( 𝐺 ‘ 𝑥 ) = ( ( 𝑔 ‘ 𝑥 ) ∪ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 68 |
67
|
3ad2ant2 |
⊢ ( ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ) ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ( 𝐺 ‘ 𝑥 ) = ( ( 𝑔 ‘ 𝑥 ) ∪ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 69 |
|
eloni |
⊢ ( 𝐴 ∈ On → Ord 𝐴 ) |
| 70 |
69
|
adantl |
⊢ ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ) → Ord 𝐴 ) |
| 71 |
70
|
3ad2ant1 |
⊢ ( ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ) ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → Ord 𝐴 ) |
| 72 |
|
f1f |
⊢ ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 → 𝑔 : ( cf ‘ 𝐴 ) ⟶ 𝐴 ) |
| 73 |
72
|
ffvelcdmda |
⊢ ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( 𝑔 ‘ 𝑥 ) ∈ 𝐴 ) |
| 74 |
73
|
adantlr |
⊢ ( ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ) ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( 𝑔 ‘ 𝑥 ) ∈ 𝐴 ) |
| 75 |
74
|
3adant3 |
⊢ ( ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ) ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ( 𝑔 ‘ 𝑥 ) ∈ 𝐴 ) |
| 76 |
2
|
fveq1i |
⊢ ( 𝐺 ‘ 𝑦 ) = ( ( recs ( 𝐹 ) ↾ ( cf ‘ 𝐴 ) ) ‘ 𝑦 ) |
| 77 |
15
|
fvresd |
⊢ ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( ( recs ( 𝐹 ) ↾ ( cf ‘ 𝐴 ) ) ‘ 𝑦 ) = ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 78 |
76 77
|
eqtrid |
⊢ ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( 𝐺 ‘ 𝑦 ) = ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 79 |
78
|
adantrl |
⊢ ( ( 𝑦 ∈ 𝑥 ∧ ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) ) → ( 𝐺 ‘ 𝑦 ) = ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 80 |
79
|
ancoms |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑥 ) → ( 𝐺 ‘ 𝑦 ) = ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 81 |
80
|
eleq1d |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑥 ) → ( ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ↔ ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 ) ) |
| 82 |
|
ordsucss |
⊢ ( Ord 𝐴 → ( ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 → suc ( recs ( 𝐹 ) ‘ 𝑦 ) ⊆ 𝐴 ) ) |
| 83 |
69 82
|
syl |
⊢ ( 𝐴 ∈ On → ( ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 → suc ( recs ( 𝐹 ) ‘ 𝑦 ) ⊆ 𝐴 ) ) |
| 84 |
83
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑥 ) → ( ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 → suc ( recs ( 𝐹 ) ‘ 𝑦 ) ⊆ 𝐴 ) ) |
| 85 |
81 84
|
sylbid |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑥 ) → ( ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 → suc ( recs ( 𝐹 ) ‘ 𝑦 ) ⊆ 𝐴 ) ) |
| 86 |
85
|
ralimdva |
⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 → ∀ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ⊆ 𝐴 ) ) |
| 87 |
|
iunss |
⊢ ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ⊆ 𝐴 ↔ ∀ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ⊆ 𝐴 ) |
| 88 |
86 87
|
imbitrrdi |
⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 → ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ⊆ 𝐴 ) ) |
| 89 |
88
|
3impia |
⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ⊆ 𝐴 ) |
| 90 |
|
onelon |
⊢ ( ( 𝐴 ∈ On ∧ ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 ) → ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ On ) |
| 91 |
90
|
ex |
⊢ ( 𝐴 ∈ On → ( ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 → ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ On ) ) |
| 92 |
91
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑥 ) → ( ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 → ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ On ) ) |
| 93 |
81 92
|
sylbid |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑥 ) → ( ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 → ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ On ) ) |
| 94 |
|
onsuc |
⊢ ( ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ On → suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ On ) |
| 95 |
93 94
|
syl6 |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) ∧ 𝑦 ∈ 𝑥 ) → ( ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 → suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ On ) ) |
| 96 |
95
|
ralimdva |
⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 → ∀ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ On ) ) |
| 97 |
96
|
3impia |
⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ∀ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ On ) |
| 98 |
|
iunon |
⊢ ( ( 𝑥 ∈ V ∧ ∀ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ On ) → ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ On ) |
| 99 |
28 97 98
|
sylancr |
⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ On ) |
| 100 |
|
simp1 |
⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → 𝐴 ∈ On ) |
| 101 |
|
onsseleq |
⊢ ( ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ On ∧ 𝐴 ∈ On ) → ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ⊆ 𝐴 ↔ ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 ∨ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ) ) ) |
| 102 |
99 100 101
|
syl2anc |
⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ⊆ 𝐴 ↔ ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 ∨ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ) ) ) |
| 103 |
|
idd |
⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 → ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 ) ) |
| 104 |
|
simpll |
⊢ ( ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ∧ ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ∧ 𝐴 ∈ On ) ) → 𝑥 ∈ ( cf ‘ 𝐴 ) ) |
| 105 |
|
simprr |
⊢ ( ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ∧ ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ∧ 𝐴 ∈ On ) ) → 𝐴 ∈ On ) |
| 106 |
5
|
ad2antrr |
⊢ ( ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ∧ ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ∧ 𝐴 ∈ On ) ) → 𝑥 ∈ On ) |
| 107 |
5 49
|
syl |
⊢ ( 𝑥 ∈ ( cf ‘ 𝐴 ) → ( recs ( 𝐹 ) ↾ 𝑥 ) Fn 𝑥 ) |
| 108 |
107
|
adantr |
⊢ ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ( recs ( 𝐹 ) ↾ 𝑥 ) Fn 𝑥 ) |
| 109 |
78
|
ancoms |
⊢ ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ 𝑦 ∈ 𝑥 ) → ( 𝐺 ‘ 𝑦 ) = ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 110 |
|
fvres |
⊢ ( 𝑦 ∈ 𝑥 → ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑦 ) = ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 111 |
110
|
adantl |
⊢ ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ 𝑦 ∈ 𝑥 ) → ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑦 ) = ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 112 |
109 111
|
eqtr4d |
⊢ ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ 𝑦 ∈ 𝑥 ) → ( 𝐺 ‘ 𝑦 ) = ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑦 ) ) |
| 113 |
112
|
eleq1d |
⊢ ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ 𝑦 ∈ 𝑥 ) → ( ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ↔ ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑦 ) ∈ 𝐴 ) ) |
| 114 |
113
|
ralbidva |
⊢ ( 𝑥 ∈ ( cf ‘ 𝐴 ) → ( ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ↔ ∀ 𝑦 ∈ 𝑥 ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑦 ) ∈ 𝐴 ) ) |
| 115 |
114
|
biimpa |
⊢ ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ∀ 𝑦 ∈ 𝑥 ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑦 ) ∈ 𝐴 ) |
| 116 |
|
ffnfv |
⊢ ( ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ↔ ( ( recs ( 𝐹 ) ↾ 𝑥 ) Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑦 ) ∈ 𝐴 ) ) |
| 117 |
108 115 116
|
sylanbrc |
⊢ ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ) |
| 118 |
|
eleq2 |
⊢ ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 → ( 𝑡 ∈ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ↔ 𝑡 ∈ 𝐴 ) ) |
| 119 |
118
|
biimpar |
⊢ ( ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ∧ 𝑡 ∈ 𝐴 ) → 𝑡 ∈ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 120 |
119
|
adantrl |
⊢ ( ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ∧ ( 𝐴 ∈ On ∧ 𝑡 ∈ 𝐴 ) ) → 𝑡 ∈ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 121 |
120
|
3adant1 |
⊢ ( ( ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ∧ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ∧ ( 𝐴 ∈ On ∧ 𝑡 ∈ 𝐴 ) ) → 𝑡 ∈ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 122 |
|
onelon |
⊢ ( ( 𝐴 ∈ On ∧ 𝑡 ∈ 𝐴 ) → 𝑡 ∈ On ) |
| 123 |
110
|
adantl |
⊢ ( ( ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ∧ 𝑦 ∈ 𝑥 ) → ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑦 ) = ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 124 |
|
ffvelcdm |
⊢ ( ( ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ∧ 𝑦 ∈ 𝑥 ) → ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑦 ) ∈ 𝐴 ) |
| 125 |
123 124
|
eqeltrrd |
⊢ ( ( ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ∧ 𝑦 ∈ 𝑥 ) → ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 ) |
| 126 |
125 90
|
sylan2 |
⊢ ( ( 𝐴 ∈ On ∧ ( ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ∧ 𝑦 ∈ 𝑥 ) ) → ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ On ) |
| 127 |
126
|
adantlr |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝑡 ∈ 𝐴 ) ∧ ( ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ∧ 𝑦 ∈ 𝑥 ) ) → ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ On ) |
| 128 |
|
onsssuc |
⊢ ( ( 𝑡 ∈ On ∧ ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ On ) → ( 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ↔ 𝑡 ∈ suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 129 |
122 127 128
|
syl2an2r |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝑡 ∈ 𝐴 ) ∧ ( ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ∧ 𝑦 ∈ 𝑥 ) ) → ( 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ↔ 𝑡 ∈ suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 130 |
129
|
anassrs |
⊢ ( ( ( ( 𝐴 ∈ On ∧ 𝑡 ∈ 𝐴 ) ∧ ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ) ∧ 𝑦 ∈ 𝑥 ) → ( 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ↔ 𝑡 ∈ suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 131 |
130
|
rexbidva |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝑡 ∈ 𝐴 ) ∧ ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ) → ( ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ↔ ∃ 𝑦 ∈ 𝑥 𝑡 ∈ suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 132 |
|
eliun |
⊢ ( 𝑡 ∈ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ↔ ∃ 𝑦 ∈ 𝑥 𝑡 ∈ suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 133 |
131 132
|
bitr4di |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝑡 ∈ 𝐴 ) ∧ ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ) → ( ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ↔ 𝑡 ∈ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 134 |
133
|
ancoms |
⊢ ( ( ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ∧ ( 𝐴 ∈ On ∧ 𝑡 ∈ 𝐴 ) ) → ( ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ↔ 𝑡 ∈ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 135 |
134
|
3adant2 |
⊢ ( ( ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ∧ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ∧ ( 𝐴 ∈ On ∧ 𝑡 ∈ 𝐴 ) ) → ( ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ↔ 𝑡 ∈ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 136 |
121 135
|
mpbird |
⊢ ( ( ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ∧ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ∧ ( 𝐴 ∈ On ∧ 𝑡 ∈ 𝐴 ) ) → ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 137 |
136
|
3expa |
⊢ ( ( ( ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ∧ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ) ∧ ( 𝐴 ∈ On ∧ 𝑡 ∈ 𝐴 ) ) → ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 138 |
137
|
anassrs |
⊢ ( ( ( ( ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ∧ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ) ∧ 𝐴 ∈ On ) ∧ 𝑡 ∈ 𝐴 ) → ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 139 |
138
|
ralrimiva |
⊢ ( ( ( ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ∧ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ) ∧ 𝐴 ∈ On ) → ∀ 𝑡 ∈ 𝐴 ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 140 |
139
|
expl |
⊢ ( ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 → ( ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ∧ 𝐴 ∈ On ) → ∀ 𝑡 ∈ 𝐴 ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 141 |
117 140
|
syl |
⊢ ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ( ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ∧ 𝐴 ∈ On ) → ∀ 𝑡 ∈ 𝐴 ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 142 |
141
|
imp |
⊢ ( ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ∧ ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ∧ 𝐴 ∈ On ) ) → ∀ 𝑡 ∈ 𝐴 ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 143 |
|
feq1 |
⊢ ( 𝑓 = ( recs ( 𝐹 ) ↾ 𝑥 ) → ( 𝑓 : 𝑥 ⟶ 𝐴 ↔ ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ) ) |
| 144 |
|
fveq1 |
⊢ ( 𝑓 = ( recs ( 𝐹 ) ↾ 𝑥 ) → ( 𝑓 ‘ 𝑦 ) = ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑦 ) ) |
| 145 |
144
|
sseq2d |
⊢ ( 𝑓 = ( recs ( 𝐹 ) ↾ 𝑥 ) → ( 𝑡 ⊆ ( 𝑓 ‘ 𝑦 ) ↔ 𝑡 ⊆ ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑦 ) ) ) |
| 146 |
145
|
rexbidv |
⊢ ( 𝑓 = ( recs ( 𝐹 ) ↾ 𝑥 ) → ( ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( 𝑓 ‘ 𝑦 ) ↔ ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑦 ) ) ) |
| 147 |
110
|
sseq2d |
⊢ ( 𝑦 ∈ 𝑥 → ( 𝑡 ⊆ ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑦 ) ↔ 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 148 |
147
|
rexbiia |
⊢ ( ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( ( recs ( 𝐹 ) ↾ 𝑥 ) ‘ 𝑦 ) ↔ ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 149 |
146 148
|
bitrdi |
⊢ ( 𝑓 = ( recs ( 𝐹 ) ↾ 𝑥 ) → ( ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( 𝑓 ‘ 𝑦 ) ↔ ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 150 |
149
|
ralbidv |
⊢ ( 𝑓 = ( recs ( 𝐹 ) ↾ 𝑥 ) → ( ∀ 𝑡 ∈ 𝐴 ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( 𝑓 ‘ 𝑦 ) ↔ ∀ 𝑡 ∈ 𝐴 ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 151 |
143 150
|
anbi12d |
⊢ ( 𝑓 = ( recs ( 𝐹 ) ↾ 𝑥 ) → ( ( 𝑓 : 𝑥 ⟶ 𝐴 ∧ ∀ 𝑡 ∈ 𝐴 ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( 𝑓 ‘ 𝑦 ) ) ↔ ( ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ∧ ∀ 𝑡 ∈ 𝐴 ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) ) |
| 152 |
30 151
|
spcev |
⊢ ( ( ( recs ( 𝐹 ) ↾ 𝑥 ) : 𝑥 ⟶ 𝐴 ∧ ∀ 𝑡 ∈ 𝐴 ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( recs ( 𝐹 ) ‘ 𝑦 ) ) → ∃ 𝑓 ( 𝑓 : 𝑥 ⟶ 𝐴 ∧ ∀ 𝑡 ∈ 𝐴 ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( 𝑓 ‘ 𝑦 ) ) ) |
| 153 |
117 142 152
|
syl2an2r |
⊢ ( ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ∧ ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ∧ 𝐴 ∈ On ) ) → ∃ 𝑓 ( 𝑓 : 𝑥 ⟶ 𝐴 ∧ ∀ 𝑡 ∈ 𝐴 ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( 𝑓 ‘ 𝑦 ) ) ) |
| 154 |
|
cfflb |
⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ On ) → ( ∃ 𝑓 ( 𝑓 : 𝑥 ⟶ 𝐴 ∧ ∀ 𝑡 ∈ 𝐴 ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( 𝑓 ‘ 𝑦 ) ) → ( cf ‘ 𝐴 ) ⊆ 𝑥 ) ) |
| 155 |
154
|
imp |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝑥 ∈ On ) ∧ ∃ 𝑓 ( 𝑓 : 𝑥 ⟶ 𝐴 ∧ ∀ 𝑡 ∈ 𝐴 ∃ 𝑦 ∈ 𝑥 𝑡 ⊆ ( 𝑓 ‘ 𝑦 ) ) ) → ( cf ‘ 𝐴 ) ⊆ 𝑥 ) |
| 156 |
105 106 153 155
|
syl21anc |
⊢ ( ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ∧ ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ∧ 𝐴 ∈ On ) ) → ( cf ‘ 𝐴 ) ⊆ 𝑥 ) |
| 157 |
|
ontri1 |
⊢ ( ( ( cf ‘ 𝐴 ) ∈ On ∧ 𝑥 ∈ On ) → ( ( cf ‘ 𝐴 ) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ( cf ‘ 𝐴 ) ) ) |
| 158 |
4 5 157
|
sylancr |
⊢ ( 𝑥 ∈ ( cf ‘ 𝐴 ) → ( ( cf ‘ 𝐴 ) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ( cf ‘ 𝐴 ) ) ) |
| 159 |
158
|
ad2antrr |
⊢ ( ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ∧ ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ∧ 𝐴 ∈ On ) ) → ( ( cf ‘ 𝐴 ) ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ( cf ‘ 𝐴 ) ) ) |
| 160 |
156 159
|
mpbid |
⊢ ( ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ∧ ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ∧ 𝐴 ∈ On ) ) → ¬ 𝑥 ∈ ( cf ‘ 𝐴 ) ) |
| 161 |
104 160
|
pm2.21dd |
⊢ ( ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ∧ ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ∧ 𝐴 ∈ On ) ) → ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 ) |
| 162 |
161
|
ex |
⊢ ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ( ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ∧ 𝐴 ∈ On ) → ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 ) ) |
| 163 |
162
|
expcomd |
⊢ ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ( 𝐴 ∈ On → ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 → ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 ) ) ) |
| 164 |
163
|
com12 |
⊢ ( 𝐴 ∈ On → ( ( 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 → ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 ) ) ) |
| 165 |
164
|
3impib |
⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 → ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 ) ) |
| 166 |
103 165
|
jaod |
⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ( ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 ∨ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) = 𝐴 ) → ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 ) ) |
| 167 |
102 166
|
sylbid |
⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ( ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ⊆ 𝐴 → ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 ) ) |
| 168 |
89 167
|
mpd |
⊢ ( ( 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 ) |
| 169 |
168
|
3adant1l |
⊢ ( ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ) ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 ) |
| 170 |
|
ordunel |
⊢ ( ( Ord 𝐴 ∧ ( 𝑔 ‘ 𝑥 ) ∈ 𝐴 ∧ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ 𝐴 ) → ( ( 𝑔 ‘ 𝑥 ) ∪ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) ∈ 𝐴 ) |
| 171 |
71 75 169 170
|
syl3anc |
⊢ ( ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ) ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ( ( 𝑔 ‘ 𝑥 ) ∪ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) ∈ 𝐴 ) |
| 172 |
68 171
|
eqeltrd |
⊢ ( ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ) ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ( 𝐺 ‘ 𝑥 ) ∈ 𝐴 ) |
| 173 |
172
|
3expia |
⊢ ( ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ) ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 → ( 𝐺 ‘ 𝑥 ) ∈ 𝐴 ) ) |
| 174 |
173
|
3impa |
⊢ ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 → ( 𝐺 ‘ 𝑥 ) ∈ 𝐴 ) ) |
| 175 |
20 174
|
syldc |
⊢ ( ∀ 𝑦 ∈ 𝑥 ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑦 ∈ ( cf ‘ 𝐴 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( 𝐺 ‘ 𝑥 ) ∈ 𝐴 ) ) |
| 176 |
175
|
a1i |
⊢ ( 𝑥 ∈ On → ( ∀ 𝑦 ∈ 𝑥 ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑦 ∈ ( cf ‘ 𝐴 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) → ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( 𝐺 ‘ 𝑥 ) ∈ 𝐴 ) ) ) |
| 177 |
11 176
|
tfis2 |
⊢ ( 𝑥 ∈ On → ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( 𝐺 ‘ 𝑥 ) ∈ 𝐴 ) ) |
| 178 |
6 177
|
mpcom |
⊢ ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( 𝐺 ‘ 𝑥 ) ∈ 𝐴 ) |
| 179 |
178
|
3expia |
⊢ ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ) → ( 𝑥 ∈ ( cf ‘ 𝐴 ) → ( 𝐺 ‘ 𝑥 ) ∈ 𝐴 ) ) |
| 180 |
179
|
ralrimiv |
⊢ ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ) → ∀ 𝑥 ∈ ( cf ‘ 𝐴 ) ( 𝐺 ‘ 𝑥 ) ∈ 𝐴 ) |
| 181 |
4
|
onssi |
⊢ ( cf ‘ 𝐴 ) ⊆ On |
| 182 |
|
fnssres |
⊢ ( ( recs ( 𝐹 ) Fn On ∧ ( cf ‘ 𝐴 ) ⊆ On ) → ( recs ( 𝐹 ) ↾ ( cf ‘ 𝐴 ) ) Fn ( cf ‘ 𝐴 ) ) |
| 183 |
2
|
fneq1i |
⊢ ( 𝐺 Fn ( cf ‘ 𝐴 ) ↔ ( recs ( 𝐹 ) ↾ ( cf ‘ 𝐴 ) ) Fn ( cf ‘ 𝐴 ) ) |
| 184 |
182 183
|
sylibr |
⊢ ( ( recs ( 𝐹 ) Fn On ∧ ( cf ‘ 𝐴 ) ⊆ On ) → 𝐺 Fn ( cf ‘ 𝐴 ) ) |
| 185 |
25 181 184
|
mp2an |
⊢ 𝐺 Fn ( cf ‘ 𝐴 ) |
| 186 |
|
ffnfv |
⊢ ( 𝐺 : ( cf ‘ 𝐴 ) ⟶ 𝐴 ↔ ( 𝐺 Fn ( cf ‘ 𝐴 ) ∧ ∀ 𝑥 ∈ ( cf ‘ 𝐴 ) ( 𝐺 ‘ 𝑥 ) ∈ 𝐴 ) ) |
| 187 |
185 186
|
mpbiran |
⊢ ( 𝐺 : ( cf ‘ 𝐴 ) ⟶ 𝐴 ↔ ∀ 𝑥 ∈ ( cf ‘ 𝐴 ) ( 𝐺 ‘ 𝑥 ) ∈ 𝐴 ) |
| 188 |
180 187
|
sylibr |
⊢ ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ) → 𝐺 : ( cf ‘ 𝐴 ) ⟶ 𝐴 ) |
| 189 |
188
|
adantlr |
⊢ ( ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝑔 ‘ 𝑤 ) ) ∧ 𝐴 ∈ On ) → 𝐺 : ( cf ‘ 𝐴 ) ⟶ 𝐴 ) |
| 190 |
|
onss |
⊢ ( 𝐴 ∈ On → 𝐴 ⊆ On ) |
| 191 |
190
|
adantl |
⊢ ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ) → 𝐴 ⊆ On ) |
| 192 |
4
|
onordi |
⊢ Ord ( cf ‘ 𝐴 ) |
| 193 |
|
fvex |
⊢ ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ V |
| 194 |
193
|
sucid |
⊢ ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ suc ( recs ( 𝐹 ) ‘ 𝑦 ) |
| 195 |
|
fveq2 |
⊢ ( 𝑡 = 𝑦 → ( recs ( 𝐹 ) ‘ 𝑡 ) = ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 196 |
|
suceq |
⊢ ( ( recs ( 𝐹 ) ‘ 𝑡 ) = ( recs ( 𝐹 ) ‘ 𝑦 ) → suc ( recs ( 𝐹 ) ‘ 𝑡 ) = suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 197 |
195 196
|
syl |
⊢ ( 𝑡 = 𝑦 → suc ( recs ( 𝐹 ) ‘ 𝑡 ) = suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 198 |
197
|
eliuni |
⊢ ( ( 𝑦 ∈ 𝑥 ∧ ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) → ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ ∪ 𝑡 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑡 ) ) |
| 199 |
198 60
|
eleqtrrdi |
⊢ ( ( 𝑦 ∈ 𝑥 ∧ ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) → ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 200 |
194 199
|
mpan2 |
⊢ ( 𝑦 ∈ 𝑥 → ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 201 |
|
elun2 |
⊢ ( ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) → ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ ( ( 𝑔 ‘ 𝑥 ) ∪ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 202 |
200 201
|
syl |
⊢ ( 𝑦 ∈ 𝑥 → ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ ( ( 𝑔 ‘ 𝑥 ) ∪ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 203 |
202
|
adantr |
⊢ ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ ( ( 𝑔 ‘ 𝑥 ) ∪ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 204 |
5
|
adantl |
⊢ ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → 𝑥 ∈ On ) |
| 205 |
204 65
|
syl |
⊢ ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( recs ( 𝐹 ) ‘ 𝑥 ) = ( ( 𝑔 ‘ 𝑥 ) ∪ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) ) |
| 206 |
203 205
|
eleqtrrd |
⊢ ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( recs ( 𝐹 ) ‘ 𝑦 ) ∈ ( recs ( 𝐹 ) ‘ 𝑥 ) ) |
| 207 |
23
|
adantl |
⊢ ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( 𝐺 ‘ 𝑥 ) = ( recs ( 𝐹 ) ‘ 𝑥 ) ) |
| 208 |
206 78 207
|
3eltr4d |
⊢ ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ( cf ‘ 𝐴 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ ( 𝐺 ‘ 𝑥 ) ) |
| 209 |
208
|
expcom |
⊢ ( 𝑥 ∈ ( cf ‘ 𝐴 ) → ( 𝑦 ∈ 𝑥 → ( 𝐺 ‘ 𝑦 ) ∈ ( 𝐺 ‘ 𝑥 ) ) ) |
| 210 |
209
|
ralrimiv |
⊢ ( 𝑥 ∈ ( cf ‘ 𝐴 ) → ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ ( 𝐺 ‘ 𝑥 ) ) |
| 211 |
210
|
rgen |
⊢ ∀ 𝑥 ∈ ( cf ‘ 𝐴 ) ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ ( 𝐺 ‘ 𝑥 ) |
| 212 |
|
issmo2 |
⊢ ( 𝐺 : ( cf ‘ 𝐴 ) ⟶ 𝐴 → ( ( 𝐴 ⊆ On ∧ Ord ( cf ‘ 𝐴 ) ∧ ∀ 𝑥 ∈ ( cf ‘ 𝐴 ) ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ ( 𝐺 ‘ 𝑥 ) ) → Smo 𝐺 ) ) |
| 213 |
212
|
com12 |
⊢ ( ( 𝐴 ⊆ On ∧ Ord ( cf ‘ 𝐴 ) ∧ ∀ 𝑥 ∈ ( cf ‘ 𝐴 ) ∀ 𝑦 ∈ 𝑥 ( 𝐺 ‘ 𝑦 ) ∈ ( 𝐺 ‘ 𝑥 ) ) → ( 𝐺 : ( cf ‘ 𝐴 ) ⟶ 𝐴 → Smo 𝐺 ) ) |
| 214 |
192 211 213
|
mp3an23 |
⊢ ( 𝐴 ⊆ On → ( 𝐺 : ( cf ‘ 𝐴 ) ⟶ 𝐴 → Smo 𝐺 ) ) |
| 215 |
191 188 214
|
sylc |
⊢ ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ 𝐴 ∈ On ) → Smo 𝐺 ) |
| 216 |
215
|
adantlr |
⊢ ( ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝑔 ‘ 𝑤 ) ) ∧ 𝐴 ∈ On ) → Smo 𝐺 ) |
| 217 |
|
fveq2 |
⊢ ( 𝑥 = 𝑤 → ( 𝑔 ‘ 𝑥 ) = ( 𝑔 ‘ 𝑤 ) ) |
| 218 |
|
fveq2 |
⊢ ( 𝑥 = 𝑤 → ( 𝐺 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑤 ) ) |
| 219 |
217 218
|
sseq12d |
⊢ ( 𝑥 = 𝑤 → ( ( 𝑔 ‘ 𝑥 ) ⊆ ( 𝐺 ‘ 𝑥 ) ↔ ( 𝑔 ‘ 𝑤 ) ⊆ ( 𝐺 ‘ 𝑤 ) ) ) |
| 220 |
|
ssun1 |
⊢ ( 𝑔 ‘ 𝑥 ) ⊆ ( ( 𝑔 ‘ 𝑥 ) ∪ ∪ 𝑦 ∈ 𝑥 suc ( recs ( 𝐹 ) ‘ 𝑦 ) ) |
| 221 |
220 67
|
sseqtrrid |
⊢ ( 𝑥 ∈ ( cf ‘ 𝐴 ) → ( 𝑔 ‘ 𝑥 ) ⊆ ( 𝐺 ‘ 𝑥 ) ) |
| 222 |
219 221
|
vtoclga |
⊢ ( 𝑤 ∈ ( cf ‘ 𝐴 ) → ( 𝑔 ‘ 𝑤 ) ⊆ ( 𝐺 ‘ 𝑤 ) ) |
| 223 |
|
sstr |
⊢ ( ( 𝑧 ⊆ ( 𝑔 ‘ 𝑤 ) ∧ ( 𝑔 ‘ 𝑤 ) ⊆ ( 𝐺 ‘ 𝑤 ) ) → 𝑧 ⊆ ( 𝐺 ‘ 𝑤 ) ) |
| 224 |
223
|
expcom |
⊢ ( ( 𝑔 ‘ 𝑤 ) ⊆ ( 𝐺 ‘ 𝑤 ) → ( 𝑧 ⊆ ( 𝑔 ‘ 𝑤 ) → 𝑧 ⊆ ( 𝐺 ‘ 𝑤 ) ) ) |
| 225 |
222 224
|
syl |
⊢ ( 𝑤 ∈ ( cf ‘ 𝐴 ) → ( 𝑧 ⊆ ( 𝑔 ‘ 𝑤 ) → 𝑧 ⊆ ( 𝐺 ‘ 𝑤 ) ) ) |
| 226 |
225
|
reximia |
⊢ ( ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝑔 ‘ 𝑤 ) → ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝐺 ‘ 𝑤 ) ) |
| 227 |
226
|
ralimi |
⊢ ( ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝑔 ‘ 𝑤 ) → ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝐺 ‘ 𝑤 ) ) |
| 228 |
227
|
ad2antlr |
⊢ ( ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝑔 ‘ 𝑤 ) ) ∧ 𝐴 ∈ On ) → ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝐺 ‘ 𝑤 ) ) |
| 229 |
|
fnex |
⊢ ( ( 𝐺 Fn ( cf ‘ 𝐴 ) ∧ ( cf ‘ 𝐴 ) ∈ On ) → 𝐺 ∈ V ) |
| 230 |
185 4 229
|
mp2an |
⊢ 𝐺 ∈ V |
| 231 |
|
feq1 |
⊢ ( 𝑓 = 𝐺 → ( 𝑓 : ( cf ‘ 𝐴 ) ⟶ 𝐴 ↔ 𝐺 : ( cf ‘ 𝐴 ) ⟶ 𝐴 ) ) |
| 232 |
|
smoeq |
⊢ ( 𝑓 = 𝐺 → ( Smo 𝑓 ↔ Smo 𝐺 ) ) |
| 233 |
|
fveq1 |
⊢ ( 𝑓 = 𝐺 → ( 𝑓 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) |
| 234 |
233
|
sseq2d |
⊢ ( 𝑓 = 𝐺 → ( 𝑧 ⊆ ( 𝑓 ‘ 𝑤 ) ↔ 𝑧 ⊆ ( 𝐺 ‘ 𝑤 ) ) ) |
| 235 |
234
|
rexbidv |
⊢ ( 𝑓 = 𝐺 → ( ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝑓 ‘ 𝑤 ) ↔ ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝐺 ‘ 𝑤 ) ) ) |
| 236 |
235
|
ralbidv |
⊢ ( 𝑓 = 𝐺 → ( ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝑓 ‘ 𝑤 ) ↔ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝐺 ‘ 𝑤 ) ) ) |
| 237 |
231 232 236
|
3anbi123d |
⊢ ( 𝑓 = 𝐺 → ( ( 𝑓 : ( cf ‘ 𝐴 ) ⟶ 𝐴 ∧ Smo 𝑓 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝑓 ‘ 𝑤 ) ) ↔ ( 𝐺 : ( cf ‘ 𝐴 ) ⟶ 𝐴 ∧ Smo 𝐺 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝐺 ‘ 𝑤 ) ) ) ) |
| 238 |
230 237
|
spcev |
⊢ ( ( 𝐺 : ( cf ‘ 𝐴 ) ⟶ 𝐴 ∧ Smo 𝐺 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝐺 ‘ 𝑤 ) ) → ∃ 𝑓 ( 𝑓 : ( cf ‘ 𝐴 ) ⟶ 𝐴 ∧ Smo 𝑓 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝑓 ‘ 𝑤 ) ) ) |
| 239 |
189 216 228 238
|
syl3anc |
⊢ ( ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝑔 ‘ 𝑤 ) ) ∧ 𝐴 ∈ On ) → ∃ 𝑓 ( 𝑓 : ( cf ‘ 𝐴 ) ⟶ 𝐴 ∧ Smo 𝑓 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝑓 ‘ 𝑤 ) ) ) |
| 240 |
239
|
expcom |
⊢ ( 𝐴 ∈ On → ( ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝑔 ‘ 𝑤 ) ) → ∃ 𝑓 ( 𝑓 : ( cf ‘ 𝐴 ) ⟶ 𝐴 ∧ Smo 𝑓 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝑓 ‘ 𝑤 ) ) ) ) |
| 241 |
240
|
exlimdv |
⊢ ( 𝐴 ∈ On → ( ∃ 𝑔 ( 𝑔 : ( cf ‘ 𝐴 ) –1-1→ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝑔 ‘ 𝑤 ) ) → ∃ 𝑓 ( 𝑓 : ( cf ‘ 𝐴 ) ⟶ 𝐴 ∧ Smo 𝑓 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝑓 ‘ 𝑤 ) ) ) ) |
| 242 |
3 241
|
mpd |
⊢ ( 𝐴 ∈ On → ∃ 𝑓 ( 𝑓 : ( cf ‘ 𝐴 ) ⟶ 𝐴 ∧ Smo 𝑓 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ ( cf ‘ 𝐴 ) 𝑧 ⊆ ( 𝑓 ‘ 𝑤 ) ) ) |