| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ttukeylem.1 |
⊢ ( 𝜑 → 𝐹 : ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) –1-1-onto→ ( ∪ 𝐴 ∖ 𝐵 ) ) |
| 2 |
|
ttukeylem.2 |
⊢ ( 𝜑 → 𝐵 ∈ 𝐴 ) |
| 3 |
|
ttukeylem.3 |
⊢ ( 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ ( 𝒫 𝑥 ∩ Fin ) ⊆ 𝐴 ) ) |
| 4 |
|
ttukeylem.4 |
⊢ 𝐺 = recs ( ( 𝑧 ∈ V ↦ if ( dom 𝑧 = ∪ dom 𝑧 , if ( dom 𝑧 = ∅ , 𝐵 , ∪ ran 𝑧 ) , ( ( 𝑧 ‘ ∪ dom 𝑧 ) ∪ if ( ( ( 𝑧 ‘ ∪ dom 𝑧 ) ∪ { ( 𝐹 ‘ ∪ dom 𝑧 ) } ) ∈ 𝐴 , { ( 𝐹 ‘ ∪ dom 𝑧 ) } , ∅ ) ) ) ) ) |
| 5 |
|
cardon |
⊢ ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∈ On |
| 6 |
5
|
onsuci |
⊢ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∈ On |
| 7 |
6
|
a1i |
⊢ ( 𝜑 → suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∈ On ) |
| 8 |
|
onelon |
⊢ ( ( suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∈ On ∧ 𝐶 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ) → 𝐶 ∈ On ) |
| 9 |
7 8
|
sylan |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ) → 𝐶 ∈ On ) |
| 10 |
|
eleq1 |
⊢ ( 𝑦 = 𝑎 → ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ↔ 𝑎 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ) ) |
| 11 |
|
fveq2 |
⊢ ( 𝑦 = 𝑎 → ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑎 ) ) |
| 12 |
11
|
eleq1d |
⊢ ( 𝑦 = 𝑎 → ( ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ↔ ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) |
| 13 |
10 12
|
imbi12d |
⊢ ( 𝑦 = 𝑎 → ( ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ↔ ( 𝑎 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ) |
| 14 |
13
|
imbi2d |
⊢ ( 𝑦 = 𝑎 → ( ( 𝜑 → ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ) ↔ ( 𝜑 → ( 𝑎 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ) ) |
| 15 |
|
eleq1 |
⊢ ( 𝑦 = 𝐶 → ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ↔ 𝐶 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ) ) |
| 16 |
|
fveq2 |
⊢ ( 𝑦 = 𝐶 → ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝐶 ) ) |
| 17 |
16
|
eleq1d |
⊢ ( 𝑦 = 𝐶 → ( ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ↔ ( 𝐺 ‘ 𝐶 ) ∈ 𝐴 ) ) |
| 18 |
15 17
|
imbi12d |
⊢ ( 𝑦 = 𝐶 → ( ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ↔ ( 𝐶 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝐶 ) ∈ 𝐴 ) ) ) |
| 19 |
18
|
imbi2d |
⊢ ( 𝑦 = 𝐶 → ( ( 𝜑 → ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ) ↔ ( 𝜑 → ( 𝐶 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝐶 ) ∈ 𝐴 ) ) ) ) |
| 20 |
|
r19.21v |
⊢ ( ∀ 𝑎 ∈ 𝑦 ( 𝜑 → ( 𝑎 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ↔ ( 𝜑 → ∀ 𝑎 ∈ 𝑦 ( 𝑎 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ) |
| 21 |
6
|
onordi |
⊢ Ord suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) |
| 22 |
21
|
a1i |
⊢ ( 𝜑 → Ord suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ) |
| 23 |
|
ordelss |
⊢ ( ( Ord suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ) → 𝑦 ⊆ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ) |
| 24 |
22 23
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ) → 𝑦 ⊆ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ) |
| 25 |
24
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ) ∧ 𝑎 ∈ 𝑦 ) → 𝑎 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ) |
| 26 |
|
biimt |
⊢ ( 𝑎 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ↔ ( 𝑎 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ) |
| 27 |
25 26
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ) ∧ 𝑎 ∈ 𝑦 ) → ( ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ↔ ( 𝑎 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ) |
| 28 |
27
|
ralbidva |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ) → ( ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ↔ ∀ 𝑎 ∈ 𝑦 ( 𝑎 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ) |
| 29 |
6
|
onssi |
⊢ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ⊆ On |
| 30 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) → 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ) |
| 31 |
29 30
|
sseldi |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) → 𝑦 ∈ On ) |
| 32 |
1 2 3 4
|
ttukeylem3 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ On ) → ( 𝐺 ‘ 𝑦 ) = if ( 𝑦 = ∪ 𝑦 , if ( 𝑦 = ∅ , 𝐵 , ∪ ( 𝐺 “ 𝑦 ) ) , ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ if ( ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ { ( 𝐹 ‘ ∪ 𝑦 ) } ) ∈ 𝐴 , { ( 𝐹 ‘ ∪ 𝑦 ) } , ∅ ) ) ) ) |
| 33 |
31 32
|
syldan |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) → ( 𝐺 ‘ 𝑦 ) = if ( 𝑦 = ∪ 𝑦 , if ( 𝑦 = ∅ , 𝐵 , ∪ ( 𝐺 “ 𝑦 ) ) , ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ if ( ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ { ( 𝐹 ‘ ∪ 𝑦 ) } ) ∈ 𝐴 , { ( 𝐹 ‘ ∪ 𝑦 ) } , ∅ ) ) ) ) |
| 34 |
2
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑦 = ∅ ) → 𝐵 ∈ 𝐴 ) |
| 35 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) → 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) |
| 36 |
35
|
elin2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) → 𝑤 ∈ Fin ) |
| 37 |
35
|
elin1d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) → 𝑤 ∈ 𝒫 ∪ ( 𝐺 “ 𝑦 ) ) |
| 38 |
37
|
elpwid |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) → 𝑤 ⊆ ∪ ( 𝐺 “ 𝑦 ) ) |
| 39 |
4
|
tfr1 |
⊢ 𝐺 Fn On |
| 40 |
|
fnfun |
⊢ ( 𝐺 Fn On → Fun 𝐺 ) |
| 41 |
|
funiunfv |
⊢ ( Fun 𝐺 → ∪ 𝑣 ∈ 𝑦 ( 𝐺 ‘ 𝑣 ) = ∪ ( 𝐺 “ 𝑦 ) ) |
| 42 |
39 40 41
|
mp2b |
⊢ ∪ 𝑣 ∈ 𝑦 ( 𝐺 ‘ 𝑣 ) = ∪ ( 𝐺 “ 𝑦 ) |
| 43 |
38 42
|
sseqtrrdi |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) → 𝑤 ⊆ ∪ 𝑣 ∈ 𝑦 ( 𝐺 ‘ 𝑣 ) ) |
| 44 |
|
dfss3 |
⊢ ( 𝑤 ⊆ ∪ 𝑣 ∈ 𝑦 ( 𝐺 ‘ 𝑣 ) ↔ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ∪ 𝑣 ∈ 𝑦 ( 𝐺 ‘ 𝑣 ) ) |
| 45 |
|
eliun |
⊢ ( 𝑢 ∈ ∪ 𝑣 ∈ 𝑦 ( 𝐺 ‘ 𝑣 ) ↔ ∃ 𝑣 ∈ 𝑦 𝑢 ∈ ( 𝐺 ‘ 𝑣 ) ) |
| 46 |
45
|
ralbii |
⊢ ( ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ∪ 𝑣 ∈ 𝑦 ( 𝐺 ‘ 𝑣 ) ↔ ∀ 𝑢 ∈ 𝑤 ∃ 𝑣 ∈ 𝑦 𝑢 ∈ ( 𝐺 ‘ 𝑣 ) ) |
| 47 |
44 46
|
bitri |
⊢ ( 𝑤 ⊆ ∪ 𝑣 ∈ 𝑦 ( 𝐺 ‘ 𝑣 ) ↔ ∀ 𝑢 ∈ 𝑤 ∃ 𝑣 ∈ 𝑦 𝑢 ∈ ( 𝐺 ‘ 𝑣 ) ) |
| 48 |
43 47
|
sylib |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) → ∀ 𝑢 ∈ 𝑤 ∃ 𝑣 ∈ 𝑦 𝑢 ∈ ( 𝐺 ‘ 𝑣 ) ) |
| 49 |
|
fveq2 |
⊢ ( 𝑣 = ( 𝑓 ‘ 𝑢 ) → ( 𝐺 ‘ 𝑣 ) = ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) |
| 50 |
49
|
eleq2d |
⊢ ( 𝑣 = ( 𝑓 ‘ 𝑢 ) → ( 𝑢 ∈ ( 𝐺 ‘ 𝑣 ) ↔ 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) |
| 51 |
50
|
ac6sfi |
⊢ ( ( 𝑤 ∈ Fin ∧ ∀ 𝑢 ∈ 𝑤 ∃ 𝑣 ∈ 𝑦 𝑢 ∈ ( 𝐺 ‘ 𝑣 ) ) → ∃ 𝑓 ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) |
| 52 |
36 48 51
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) → ∃ 𝑓 ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) |
| 53 |
|
eleq1 |
⊢ ( 𝑤 = ∅ → ( 𝑤 ∈ 𝐴 ↔ ∅ ∈ 𝐴 ) ) |
| 54 |
|
simp-4l |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) ∧ 𝑤 ≠ ∅ ) → 𝜑 ) |
| 55 |
|
fveq2 |
⊢ ( 𝑎 = ∪ ran 𝑓 → ( 𝐺 ‘ 𝑎 ) = ( 𝐺 ‘ ∪ ran 𝑓 ) ) |
| 56 |
55
|
eleq1d |
⊢ ( 𝑎 = ∪ ran 𝑓 → ( ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ↔ ( 𝐺 ‘ ∪ ran 𝑓 ) ∈ 𝐴 ) ) |
| 57 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) → ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) |
| 58 |
57
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) ∧ 𝑤 ≠ ∅ ) → ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) |
| 59 |
|
simprrl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) → 𝑓 : 𝑤 ⟶ 𝑦 ) |
| 60 |
59
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) ∧ 𝑤 ≠ ∅ ) → 𝑓 : 𝑤 ⟶ 𝑦 ) |
| 61 |
|
frn |
⊢ ( 𝑓 : 𝑤 ⟶ 𝑦 → ran 𝑓 ⊆ 𝑦 ) |
| 62 |
60 61
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) ∧ 𝑤 ≠ ∅ ) → ran 𝑓 ⊆ 𝑦 ) |
| 63 |
31
|
ad3antrrr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) ∧ 𝑤 ≠ ∅ ) → 𝑦 ∈ On ) |
| 64 |
|
onss |
⊢ ( 𝑦 ∈ On → 𝑦 ⊆ On ) |
| 65 |
63 64
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) ∧ 𝑤 ≠ ∅ ) → 𝑦 ⊆ On ) |
| 66 |
62 65
|
sstrd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) ∧ 𝑤 ≠ ∅ ) → ran 𝑓 ⊆ On ) |
| 67 |
36
|
adantrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) → 𝑤 ∈ Fin ) |
| 68 |
67
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) ∧ 𝑤 ≠ ∅ ) → 𝑤 ∈ Fin ) |
| 69 |
|
ffn |
⊢ ( 𝑓 : 𝑤 ⟶ 𝑦 → 𝑓 Fn 𝑤 ) |
| 70 |
60 69
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) ∧ 𝑤 ≠ ∅ ) → 𝑓 Fn 𝑤 ) |
| 71 |
|
dffn4 |
⊢ ( 𝑓 Fn 𝑤 ↔ 𝑓 : 𝑤 –onto→ ran 𝑓 ) |
| 72 |
70 71
|
sylib |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) ∧ 𝑤 ≠ ∅ ) → 𝑓 : 𝑤 –onto→ ran 𝑓 ) |
| 73 |
|
fofi |
⊢ ( ( 𝑤 ∈ Fin ∧ 𝑓 : 𝑤 –onto→ ran 𝑓 ) → ran 𝑓 ∈ Fin ) |
| 74 |
68 72 73
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) ∧ 𝑤 ≠ ∅ ) → ran 𝑓 ∈ Fin ) |
| 75 |
|
dm0rn0 |
⊢ ( dom 𝑓 = ∅ ↔ ran 𝑓 = ∅ ) |
| 76 |
59
|
fdmd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) → dom 𝑓 = 𝑤 ) |
| 77 |
76
|
eqeq1d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) → ( dom 𝑓 = ∅ ↔ 𝑤 = ∅ ) ) |
| 78 |
75 77
|
syl5bbr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) → ( ran 𝑓 = ∅ ↔ 𝑤 = ∅ ) ) |
| 79 |
78
|
necon3bid |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) → ( ran 𝑓 ≠ ∅ ↔ 𝑤 ≠ ∅ ) ) |
| 80 |
79
|
biimpar |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) ∧ 𝑤 ≠ ∅ ) → ran 𝑓 ≠ ∅ ) |
| 81 |
|
ordunifi |
⊢ ( ( ran 𝑓 ⊆ On ∧ ran 𝑓 ∈ Fin ∧ ran 𝑓 ≠ ∅ ) → ∪ ran 𝑓 ∈ ran 𝑓 ) |
| 82 |
66 74 80 81
|
syl3anc |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) ∧ 𝑤 ≠ ∅ ) → ∪ ran 𝑓 ∈ ran 𝑓 ) |
| 83 |
62 82
|
sseldd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) ∧ 𝑤 ≠ ∅ ) → ∪ ran 𝑓 ∈ 𝑦 ) |
| 84 |
56 58 83
|
rspcdva |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) ∧ 𝑤 ≠ ∅ ) → ( 𝐺 ‘ ∪ ran 𝑓 ) ∈ 𝐴 ) |
| 85 |
|
simp-4l |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ 𝑢 ∈ 𝑤 ) ) → 𝜑 ) |
| 86 |
31
|
ad3antrrr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ 𝑢 ∈ 𝑤 ) ) → 𝑦 ∈ On ) |
| 87 |
86 64
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ 𝑢 ∈ 𝑤 ) ) → 𝑦 ⊆ On ) |
| 88 |
|
ffvelrn |
⊢ ( ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ 𝑢 ∈ 𝑤 ) → ( 𝑓 ‘ 𝑢 ) ∈ 𝑦 ) |
| 89 |
88
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ 𝑢 ∈ 𝑤 ) ) → ( 𝑓 ‘ 𝑢 ) ∈ 𝑦 ) |
| 90 |
87 89
|
sseldd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ 𝑢 ∈ 𝑤 ) ) → ( 𝑓 ‘ 𝑢 ) ∈ On ) |
| 91 |
61
|
ad2antrl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ 𝑢 ∈ 𝑤 ) ) → ran 𝑓 ⊆ 𝑦 ) |
| 92 |
91 87
|
sstrd |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ 𝑢 ∈ 𝑤 ) ) → ran 𝑓 ⊆ On ) |
| 93 |
|
vex |
⊢ 𝑓 ∈ V |
| 94 |
93
|
rnex |
⊢ ran 𝑓 ∈ V |
| 95 |
94
|
ssonunii |
⊢ ( ran 𝑓 ⊆ On → ∪ ran 𝑓 ∈ On ) |
| 96 |
92 95
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ 𝑢 ∈ 𝑤 ) ) → ∪ ran 𝑓 ∈ On ) |
| 97 |
69
|
ad2antrl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ 𝑢 ∈ 𝑤 ) ) → 𝑓 Fn 𝑤 ) |
| 98 |
|
simprr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ 𝑢 ∈ 𝑤 ) ) → 𝑢 ∈ 𝑤 ) |
| 99 |
|
fnfvelrn |
⊢ ( ( 𝑓 Fn 𝑤 ∧ 𝑢 ∈ 𝑤 ) → ( 𝑓 ‘ 𝑢 ) ∈ ran 𝑓 ) |
| 100 |
97 98 99
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ 𝑢 ∈ 𝑤 ) ) → ( 𝑓 ‘ 𝑢 ) ∈ ran 𝑓 ) |
| 101 |
|
elssuni |
⊢ ( ( 𝑓 ‘ 𝑢 ) ∈ ran 𝑓 → ( 𝑓 ‘ 𝑢 ) ⊆ ∪ ran 𝑓 ) |
| 102 |
100 101
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ 𝑢 ∈ 𝑤 ) ) → ( 𝑓 ‘ 𝑢 ) ⊆ ∪ ran 𝑓 ) |
| 103 |
1 2 3 4
|
ttukeylem5 |
⊢ ( ( 𝜑 ∧ ( ( 𝑓 ‘ 𝑢 ) ∈ On ∧ ∪ ran 𝑓 ∈ On ∧ ( 𝑓 ‘ 𝑢 ) ⊆ ∪ ran 𝑓 ) ) → ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ⊆ ( 𝐺 ‘ ∪ ran 𝑓 ) ) |
| 104 |
85 90 96 102 103
|
syl13anc |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ 𝑢 ∈ 𝑤 ) ) → ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ⊆ ( 𝐺 ‘ ∪ ran 𝑓 ) ) |
| 105 |
104
|
sseld |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ 𝑢 ∈ 𝑤 ) ) → ( 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) → 𝑢 ∈ ( 𝐺 ‘ ∪ ran 𝑓 ) ) ) |
| 106 |
105
|
anassrs |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) ∧ 𝑓 : 𝑤 ⟶ 𝑦 ) ∧ 𝑢 ∈ 𝑤 ) → ( 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) → 𝑢 ∈ ( 𝐺 ‘ ∪ ran 𝑓 ) ) ) |
| 107 |
106
|
ralimdva |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) ∧ 𝑓 : 𝑤 ⟶ 𝑦 ) → ( ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) → ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ∪ ran 𝑓 ) ) ) |
| 108 |
107
|
expimpd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) → ( ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) → ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ∪ ran 𝑓 ) ) ) |
| 109 |
108
|
impr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) → ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ∪ ran 𝑓 ) ) |
| 110 |
109
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) ∧ 𝑤 ≠ ∅ ) → ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ∪ ran 𝑓 ) ) |
| 111 |
|
dfss3 |
⊢ ( 𝑤 ⊆ ( 𝐺 ‘ ∪ ran 𝑓 ) ↔ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ∪ ran 𝑓 ) ) |
| 112 |
110 111
|
sylibr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) ∧ 𝑤 ≠ ∅ ) → 𝑤 ⊆ ( 𝐺 ‘ ∪ ran 𝑓 ) ) |
| 113 |
1 2 3
|
ttukeylem2 |
⊢ ( ( 𝜑 ∧ ( ( 𝐺 ‘ ∪ ran 𝑓 ) ∈ 𝐴 ∧ 𝑤 ⊆ ( 𝐺 ‘ ∪ ran 𝑓 ) ) ) → 𝑤 ∈ 𝐴 ) |
| 114 |
54 84 112 113
|
syl12anc |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) ∧ 𝑤 ≠ ∅ ) → 𝑤 ∈ 𝐴 ) |
| 115 |
|
0ss |
⊢ ∅ ⊆ 𝐵 |
| 116 |
1 2 3
|
ttukeylem2 |
⊢ ( ( 𝜑 ∧ ( 𝐵 ∈ 𝐴 ∧ ∅ ⊆ 𝐵 ) ) → ∅ ∈ 𝐴 ) |
| 117 |
115 116
|
mpanr2 |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ 𝐴 ) → ∅ ∈ 𝐴 ) |
| 118 |
2 117
|
mpdan |
⊢ ( 𝜑 → ∅ ∈ 𝐴 ) |
| 119 |
118
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) → ∅ ∈ 𝐴 ) |
| 120 |
53 114 119
|
pm2.61ne |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ∧ ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) ) ) → 𝑤 ∈ 𝐴 ) |
| 121 |
120
|
expr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) → ( ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) → 𝑤 ∈ 𝐴 ) ) |
| 122 |
121
|
exlimdv |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) → ( ∃ 𝑓 ( 𝑓 : 𝑤 ⟶ 𝑦 ∧ ∀ 𝑢 ∈ 𝑤 𝑢 ∈ ( 𝐺 ‘ ( 𝑓 ‘ 𝑢 ) ) ) → 𝑤 ∈ 𝐴 ) ) |
| 123 |
52 122
|
mpd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ) → 𝑤 ∈ 𝐴 ) |
| 124 |
123
|
ex |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) → ( 𝑤 ∈ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) → 𝑤 ∈ 𝐴 ) ) |
| 125 |
124
|
ssrdv |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) → ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ⊆ 𝐴 ) |
| 126 |
1 2 3
|
ttukeylem1 |
⊢ ( 𝜑 → ( ∪ ( 𝐺 “ 𝑦 ) ∈ 𝐴 ↔ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ⊆ 𝐴 ) ) |
| 127 |
126
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) → ( ∪ ( 𝐺 “ 𝑦 ) ∈ 𝐴 ↔ ( 𝒫 ∪ ( 𝐺 “ 𝑦 ) ∩ Fin ) ⊆ 𝐴 ) ) |
| 128 |
125 127
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) → ∪ ( 𝐺 “ 𝑦 ) ∈ 𝐴 ) |
| 129 |
128
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) ∧ ¬ 𝑦 = ∅ ) → ∪ ( 𝐺 “ 𝑦 ) ∈ 𝐴 ) |
| 130 |
34 129
|
ifclda |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ 𝑦 = ∪ 𝑦 ) → if ( 𝑦 = ∅ , 𝐵 , ∪ ( 𝐺 “ 𝑦 ) ) ∈ 𝐴 ) |
| 131 |
|
uneq2 |
⊢ ( { ( 𝐹 ‘ ∪ 𝑦 ) } = if ( ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ { ( 𝐹 ‘ ∪ 𝑦 ) } ) ∈ 𝐴 , { ( 𝐹 ‘ ∪ 𝑦 ) } , ∅ ) → ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ { ( 𝐹 ‘ ∪ 𝑦 ) } ) = ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ if ( ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ { ( 𝐹 ‘ ∪ 𝑦 ) } ) ∈ 𝐴 , { ( 𝐹 ‘ ∪ 𝑦 ) } , ∅ ) ) ) |
| 132 |
131
|
eleq1d |
⊢ ( { ( 𝐹 ‘ ∪ 𝑦 ) } = if ( ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ { ( 𝐹 ‘ ∪ 𝑦 ) } ) ∈ 𝐴 , { ( 𝐹 ‘ ∪ 𝑦 ) } , ∅ ) → ( ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ { ( 𝐹 ‘ ∪ 𝑦 ) } ) ∈ 𝐴 ↔ ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ if ( ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ { ( 𝐹 ‘ ∪ 𝑦 ) } ) ∈ 𝐴 , { ( 𝐹 ‘ ∪ 𝑦 ) } , ∅ ) ) ∈ 𝐴 ) ) |
| 133 |
|
un0 |
⊢ ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ ∅ ) = ( 𝐺 ‘ ∪ 𝑦 ) |
| 134 |
|
uneq2 |
⊢ ( ∅ = if ( ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ { ( 𝐹 ‘ ∪ 𝑦 ) } ) ∈ 𝐴 , { ( 𝐹 ‘ ∪ 𝑦 ) } , ∅ ) → ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ ∅ ) = ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ if ( ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ { ( 𝐹 ‘ ∪ 𝑦 ) } ) ∈ 𝐴 , { ( 𝐹 ‘ ∪ 𝑦 ) } , ∅ ) ) ) |
| 135 |
133 134
|
syl5eqr |
⊢ ( ∅ = if ( ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ { ( 𝐹 ‘ ∪ 𝑦 ) } ) ∈ 𝐴 , { ( 𝐹 ‘ ∪ 𝑦 ) } , ∅ ) → ( 𝐺 ‘ ∪ 𝑦 ) = ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ if ( ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ { ( 𝐹 ‘ ∪ 𝑦 ) } ) ∈ 𝐴 , { ( 𝐹 ‘ ∪ 𝑦 ) } , ∅ ) ) ) |
| 136 |
135
|
eleq1d |
⊢ ( ∅ = if ( ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ { ( 𝐹 ‘ ∪ 𝑦 ) } ) ∈ 𝐴 , { ( 𝐹 ‘ ∪ 𝑦 ) } , ∅ ) → ( ( 𝐺 ‘ ∪ 𝑦 ) ∈ 𝐴 ↔ ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ if ( ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ { ( 𝐹 ‘ ∪ 𝑦 ) } ) ∈ 𝐴 , { ( 𝐹 ‘ ∪ 𝑦 ) } , ∅ ) ) ∈ 𝐴 ) ) |
| 137 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ ¬ 𝑦 = ∪ 𝑦 ) ∧ ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ { ( 𝐹 ‘ ∪ 𝑦 ) } ) ∈ 𝐴 ) → ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ { ( 𝐹 ‘ ∪ 𝑦 ) } ) ∈ 𝐴 ) |
| 138 |
|
fveq2 |
⊢ ( 𝑎 = ∪ 𝑦 → ( 𝐺 ‘ 𝑎 ) = ( 𝐺 ‘ ∪ 𝑦 ) ) |
| 139 |
138
|
eleq1d |
⊢ ( 𝑎 = ∪ 𝑦 → ( ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ↔ ( 𝐺 ‘ ∪ 𝑦 ) ∈ 𝐴 ) ) |
| 140 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ ¬ 𝑦 = ∪ 𝑦 ) → ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) |
| 141 |
|
vuniex |
⊢ ∪ 𝑦 ∈ V |
| 142 |
141
|
sucid |
⊢ ∪ 𝑦 ∈ suc ∪ 𝑦 |
| 143 |
|
eloni |
⊢ ( 𝑦 ∈ On → Ord 𝑦 ) |
| 144 |
|
orduniorsuc |
⊢ ( Ord 𝑦 → ( 𝑦 = ∪ 𝑦 ∨ 𝑦 = suc ∪ 𝑦 ) ) |
| 145 |
31 143 144
|
3syl |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) → ( 𝑦 = ∪ 𝑦 ∨ 𝑦 = suc ∪ 𝑦 ) ) |
| 146 |
145
|
orcanai |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ ¬ 𝑦 = ∪ 𝑦 ) → 𝑦 = suc ∪ 𝑦 ) |
| 147 |
142 146
|
eleqtrrid |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ ¬ 𝑦 = ∪ 𝑦 ) → ∪ 𝑦 ∈ 𝑦 ) |
| 148 |
139 140 147
|
rspcdva |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ ¬ 𝑦 = ∪ 𝑦 ) → ( 𝐺 ‘ ∪ 𝑦 ) ∈ 𝐴 ) |
| 149 |
148
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ ¬ 𝑦 = ∪ 𝑦 ) ∧ ¬ ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ { ( 𝐹 ‘ ∪ 𝑦 ) } ) ∈ 𝐴 ) → ( 𝐺 ‘ ∪ 𝑦 ) ∈ 𝐴 ) |
| 150 |
132 136 137 149
|
ifbothda |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) ∧ ¬ 𝑦 = ∪ 𝑦 ) → ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ if ( ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ { ( 𝐹 ‘ ∪ 𝑦 ) } ) ∈ 𝐴 , { ( 𝐹 ‘ ∪ 𝑦 ) } , ∅ ) ) ∈ 𝐴 ) |
| 151 |
130 150
|
ifclda |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) → if ( 𝑦 = ∪ 𝑦 , if ( 𝑦 = ∅ , 𝐵 , ∪ ( 𝐺 “ 𝑦 ) ) , ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ if ( ( ( 𝐺 ‘ ∪ 𝑦 ) ∪ { ( 𝐹 ‘ ∪ 𝑦 ) } ) ∈ 𝐴 , { ( 𝐹 ‘ ∪ 𝑦 ) } , ∅ ) ) ) ∈ 𝐴 ) |
| 152 |
33 151
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ∧ ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) |
| 153 |
152
|
expr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ) → ( ∀ 𝑎 ∈ 𝑦 ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ) |
| 154 |
28 153
|
sylbird |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ) → ( ∀ 𝑎 ∈ 𝑦 ( 𝑎 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ) |
| 155 |
154
|
ex |
⊢ ( 𝜑 → ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( ∀ 𝑎 ∈ 𝑦 ( 𝑎 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ) ) |
| 156 |
155
|
com23 |
⊢ ( 𝜑 → ( ∀ 𝑎 ∈ 𝑦 ( 𝑎 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) → ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ) ) |
| 157 |
156
|
a2i |
⊢ ( ( 𝜑 → ∀ 𝑎 ∈ 𝑦 ( 𝑎 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) → ( 𝜑 → ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ) ) |
| 158 |
20 157
|
sylbi |
⊢ ( ∀ 𝑎 ∈ 𝑦 ( 𝜑 → ( 𝑎 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) → ( 𝜑 → ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ) ) |
| 159 |
158
|
a1i |
⊢ ( 𝑦 ∈ On → ( ∀ 𝑎 ∈ 𝑦 ( 𝜑 → ( 𝑎 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑎 ) ∈ 𝐴 ) ) → ( 𝜑 → ( 𝑦 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝐴 ) ) ) ) |
| 160 |
14 19 159
|
tfis3 |
⊢ ( 𝐶 ∈ On → ( 𝜑 → ( 𝐶 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) → ( 𝐺 ‘ 𝐶 ) ∈ 𝐴 ) ) ) |
| 161 |
160
|
impd |
⊢ ( 𝐶 ∈ On → ( ( 𝜑 ∧ 𝐶 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ) → ( 𝐺 ‘ 𝐶 ) ∈ 𝐴 ) ) |
| 162 |
9 161
|
mpcom |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ suc ( card ‘ ( ∪ 𝐴 ∖ 𝐵 ) ) ) → ( 𝐺 ‘ 𝐶 ) ∈ 𝐴 ) |